formal/mizar/aimloop.miz

:: {AIM } Loops and the {AIM } Conjecture
::  by Chad E. Brown and Karol P\kak

environ

 vocabularies STRUCT_0, ALGSTR_0, BINOP_1, SUBSET_1, SETFAM_1, ABIAN, KNASTER,
      FUNCT_1, RELAT_1, XBOOLE_0, ALGSTR_1, ZFMISC_1, GROUP_6, GROUP_9,
      MESFUNC1, VECTSP_1, TARSKI, REALSET1, COHSP_1, ARYTM_3, FUNCT_2,
      PRE_TOPC, QC_LANG1, AUTGROUP, AIMLOOP, FUNCT_5, GROUP_1, FUNCOP_1;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, REALSET1, RELAT_1, FUNCT_1,
      PARTFUN1, FUNCT_2, BINOP_1, FUNCOP_1, STRUCT_0, ALGSTR_0, ALGSTR_1,
      GROUP_1, VECTSP_1, ABIAN, KNASTER, FUNCT_5;
 constructors BINOP_2, ALGSTR_1, REALSET1, VECTSP_2, GR_CY_1, RELSET_1, ABIAN,
      KNASTER, FUNCT_5;
 registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, KNASTER, PARTFUN1,
      FUNCT_2, REALSET1, STRUCT_0, ALGSTR_0, ALGSTR_1, VECTSP_1, RELSET_1,
      FUNCOP_1;
 requirements BOOLE, SUBSET;
 definitions TARSKI, XBOOLE_0, FUNCT_2, GROUP_1, COHSP_1, ALGSTR_0, ALGSTR_1;
 equalities BINOP_1, STRUCT_0, REALSET1, ALGSTR_0;
 expansions XBOOLE_0, TARSKI, BINOP_1, VECTSP_1;
 theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, KNASTER, ZFMISC_1, FUNCT_1,
      FUNCT_2, RELAT_1, ALGSTR_0, ALGSTR_1, REALSET1, ABIAN, RING_3, SYSREL,
      FUNCT_5;
 schemes FUNCT_2, FUNCT_7, SUBSET_1;

begin :: Loops - Introduction

::We define division operations on loops,
::inner mappings T, L and R, commutators and associators
::and basic attributes of interest. We also consider
::subloops and homomorphisms. Particular subloops are
::the nucleus and center of a loop and kernels of homomorphisms.

reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

definition
  let X be 1-sorted;
  mode Permutation of X is Permutation of the carrier of X;
  let Y be 1-sorted;
  func Funcs(X,Y) -> set equals
    Funcs(the carrier of X,the carrier of Y);
  coherence;
end;

registration
  let X,Y be 1-sorted;
  cluster Funcs(X,Y) -> functional;
  coherence;
end;

definition
  let Q be invertible left_mult-cancelable non empty multLoopStr,
      x,y be Element of Q;
  func x \ y -> Element of Q means :Def2:
    x * it = y;
  existence by ALGSTR_1:def 6;
  uniqueness by ALGSTR_0:def 20;
end;

definition
  let Q be invertible right_mult-cancelable non empty multLoopStr,
      x,y be Element of Q;
  func x / y -> Element of Q means :Def3:
    it * y = x;
  existence by ALGSTR_1:def 6;
  uniqueness by ALGSTR_0:def 21;
end;

registration
  let Q,x,y;
  reduce x \ (x * y) to y;
  reducibility by Def2;
  reduce x * (x \ y) to y;
  reducibility by Def2;
  reduce (x * y) / y to x;
  reducibility by Def3;
  reduce (x / y) * y to x;
  reducibility by Def3;
end;

definition
  let Q be invertible left_mult-cancelable non empty multLoopStr,
      u,x be Element of Q;
  func T_map(u,x) -> Element of Q equals
    x \ (u * x);
  coherence;
end;

definition
  let Q be invertible left_mult-cancelable non empty multLoopStr,
      u,x,y be Element of Q;
  func L_map(u,x,y) -> Element of Q equals
    (y * x) \ (y * (x * u));
  coherence;
end;

definition
  let Q be invertible right_mult-cancelable non empty multLoopStr,
    u,x,y be Element of Q;
  func R_map(u,x,y) -> Element of Q equals
    ((u * x) * y) / (x * y);
  coherence;
end;

definition
  let Q;
  attr Q is satisfying_TT means
    for u,x,y be Element of Q holds
      T_map(T_map(u,x),y) = T_map(T_map(u,y),x);
  attr Q is satisfying_TL means
    for u,x,y,z be Element of Q holds
      T_map(L_map(u,x,y),z) = L_map(T_map(u,z),x,y);
  attr Q is satisfying_TR means
    for u,x,y,z be Element of Q holds
      T_map(R_map(u,x,y),z) = R_map(T_map(u,z),x,y);
  attr Q is satisfying_LR means
    for u,x,y,z,w be Element of Q holds
      L_map(R_map(u,x,y),z,w) = R_map(L_map(u,z,w),x,y);
  attr Q is satisfying_LL means
    for u,x,y,z,w be Element of Q holds
      L_map(L_map(u,x,y),z,w) = L_map(L_map(u,z,w),x,y);
  attr Q is satisfying_RR means
    for u,x,y,z,w be Element of Q holds
      R_map(R_map(u,x,y),z,w) = R_map(R_map(u,z,w),x,y);
end;

definition
  let Q,x,y;
  func K_op(x,y) -> Element of Q equals
    (y * x) \ (x * y);
  coherence;
end;

definition
  let Q,x,y,z;
  func a_op(x,y,z) -> Element of Q equals
    (x * (y * z)) \ ((x * y) * z);
  coherence;
end;

definition
  let Q be multLoop;
  attr Q is satisfying_aa1 means :Def15:
    for x,y,z,u,w be Element of Q holds a_op(a_op(x,y,z),u,w) = 1.Q;
  attr Q is satisfying_aa2 means :Def16:
    for x,y,z,u,w be Element of Q holds a_op(x,a_op(y,z,u),w) = 1.Q;
  attr Q is satisfying_aa3 means :Def17:
    for x,y,z,u,w be Element of Q holds a_op(x,y,a_op(z,u,w)) = 1.Q;
  attr Q is satisfying_Ka means :Def18:
    for x,y,z,u be Element of Q holds K_op(a_op(x,y,z),u) = 1.Q;
  attr Q is satisfying_aK1 means :Def19:
    for x,y,z,u be Element of Q holds a_op(K_op(x,y),z,u) = 1.Q;
  attr Q is satisfying_aK2 means :Def20:
    for x,y,z,u be Element of Q holds a_op(x,K_op(y,z),u) = 1.Q;
  attr Q is satisfying_aK3 means :Def21:
    for x,y,z,u be Element of Q holds a_op(x,y,K_op(z,u)) = 1.Q;
end;

registration
  cluster strict satisfying_TT satisfying_TL satisfying_TR satisfying_LR
                 satisfying_LL satisfying_RR
                 satisfying_aa1 satisfying_aa2 satisfying_aa3
                 satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3
    for multLoop;
  existence
  proof
    Trivial-multLoopStr is satisfying_TT satisfying_TL satisfying_TR
       satisfying_LR satisfying_LL satisfying_RR
       satisfying_aa1 satisfying_aa2 satisfying_aa3
      satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3 by ALGSTR_1:9;
    hence thesis;
  end;
end;

theorem Th1:
  x * y = u & x * z = u implies y = z
proof
  assume x * y = u & x * z = u;
  then x \ (x * y) = x \ (x * z);
  hence thesis;
end;

theorem Th2:
  y * x = u & z * x = u implies y = z
proof
  assume y * x = u & z * x = u;
  then (y * x) / x = (z * x) / x;
  hence thesis;
end;

theorem
  x * y = x * z implies y = z by Th1;

theorem
  y * x = z * x implies y = z by Th2;

registration
  let Q,x;
  reduce 1.Q \ x to x;
  reducibility
  proof
    1.Q * x = x;
    hence thesis;
  end;
  reduce x / 1.Q to x;
  reducibility
  proof
    x * 1.Q = x;
    hence thesis;
  end;
  let y;
  reduce y / (x \ y) to x;
  reducibility
  proof
    x * (x \ y) = y;
    hence thesis;
  end;
  reduce (y / x) \ y to x;
  reducibility
  proof
    (y / x) * x = y;
    hence thesis;
  end;
end;

theorem Th5:
  x \ x = 1.Q
proof
  x * 1.Q = x;
  hence thesis;
end;

theorem Th6:
  x / x = 1.Q
proof
  1.Q * x = x;
  hence thesis;
end;

theorem
  x \ y = 1.Q implies x = y
proof
  assume x \ y = 1.Q;
  then x * 1.Q = y;
  hence thesis;
end;

theorem
  x / y = 1.Q implies x = y
proof
  assume x / y = 1.Q;
  then 1.Q * y = x;
  hence x = y;
end;

theorem Th9:
  a_op(x,y,z) = 1.Q implies x*(y*z) = (x*y)*z
proof
  assume a_op(x,y,z) = 1.Q;
  then (x*(y*z)) * 1.Q = ((x*y)*z);
  hence thesis;
end;

theorem Th10:
  K_op(x,y) = 1.Q implies x*y = y*x
proof
  assume K_op(x,y) = 1.Q;
  then (y*x) * 1.Q = x*y;
  hence thesis;
end;

theorem
  a_op(x,y,z) = 1.Q implies L_map(z,y,x) = z
proof
  assume a_op(x,y,z) = 1.Q;
  then L_map(z,y,x) = (x*y) \ ((x*y)*z) by Th9;
  hence thesis;
end;

definition
  let Q;
  defpred P1[Element of Q] means
    for y,z holds ($1 * y) * z = $1 * (y * z);
  defpred P2[Element of Q] means
    for x,z holds (x * $1) * z = x * ($1 * z);
  defpred P3[Element of Q] means
    for x,y holds (x * y) * $1 = x * (y * $1);
  defpred PC[Element of Q] means
    for y holds $1 * y = y * $1;
  func Nucl_l Q -> Subset of Q means :Def22:
    x in it iff for y,z holds (x * y) * z = x * (y * z);
  existence
  proof
    set N = {x : P1[x]};
    N c= the carrier of Q
    proof
      let x be object;
      assume x in N;
      then ex x1 be Element of Q st x = x1 & P1[x1];
      hence thesis;
    end;
    then reconsider N as Subset of Q;
    take N;
    let x;
    now
      assume x in N;
      then ex x1 be Element of Q st x = x1 & P1[x1];
      hence P1[x];
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let X1,X2 be Subset of Q such that
     A1: for x being Element of Q holds x in X1 iff P1[x]
    and A2: for x being Element of Q holds x in X2 iff P1[x];
    thus thesis from SUBSET_1:sch 2(A1,A2);
  end;
  func Nucl_m Q -> Subset of Q means
  :Def23:
  y in it iff for x,z holds (x * y) * z = x * (y * z);
  existence
  proof
    set N = {x : P2[x]};
    N c= the carrier of Q
    proof
      let x be object;
      assume x in N;
      then ex x1 be Element of Q st x = x1 & P2[x1];
      hence thesis;
    end;
    then reconsider N as Subset of Q;
    take N;
    let x;
    now
      assume x in N;
      then ex x1 be Element of Q st
      x = x1 & P2[x1];
      hence P2[x];
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let X1,X2 be Subset of Q such that
     A3: for x being Element of Q holds x in X1 iff P2[x] and
     A4: for x being Element of Q holds x in X2 iff P2[x];
    thus thesis from SUBSET_1:sch 2(A3,A4);
  end;
  func Nucl_r Q -> Subset of Q means
  :Def24:
  z in it iff for x,y holds (x * y) * z = x * (y * z);
  existence
  proof
    set N = {x : P3[x]};
    N c= the carrier of Q
    proof
      let x be object;
      assume x in N;
      then ex x1 be Element of Q st x = x1 & P3[x1];
      hence thesis;
    end;
    then reconsider N as Subset of Q;
    take N;
    let x;
    x in N implies P3[x]
    proof
      assume x in N;
      then ex x1 be Element of Q st x = x1 & P3[x1];
      hence thesis;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let X1,X2 be Subset of Q such that
        A6: for x being Element of Q holds x in X1 iff P3[x]
    and A7: for x being Element of Q holds x in X2 iff P3[x];
    thus thesis from SUBSET_1:sch 2(A6,A7);
  end;
  func Comm Q -> Subset of Q means
  :Def25:
  x in it iff for y holds x * y = y * x;
  existence
  proof
    set N = {x : PC[x]};
    N c= the carrier of Q
    proof
      let x be object;
      assume x in N;
      then ex x1 be Element of Q st x = x1 & PC[x1];
      hence thesis;
    end;
    then reconsider N as Subset of Q;
    take N;
    let x;
    x in N implies PC[x]
    proof
      assume x in N;
      then ex x1 be Element of Q st x = x1 & PC[x1];
      hence thesis;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let X1,X2 be Subset of Q such that
        A9: for x being Element of Q holds x in X1 iff PC[x]
    and A10: for x being Element of Q holds x in X2 iff PC[x];
    thus thesis from SUBSET_1:sch 2(A9,A10);
  end;
end;

definition
  let Q;
  func Nucl Q -> Subset of Q equals
    Nucl_l Q /\ Nucl_m Q /\ Nucl_r Q;
  coherence;
end;

theorem Th12:
  x in Nucl Q iff x in Nucl_l Q & x in Nucl_m Q & x in Nucl_r Q
proof
  thus x in Nucl Q implies x in Nucl_l Q & x in Nucl_m Q & x in Nucl_r Q
  proof
    assume A1: x in Nucl Q;
    then x in Nucl_l Q /\ Nucl_m Q by XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 4, A1;
  end;
  assume that A2: x in Nucl_l Q & x in Nucl_m Q
  and A3: x in Nucl_r Q;
  x in Nucl_l Q /\ Nucl_m Q by XBOOLE_0:def 4, A2;
  hence x in Nucl Q by XBOOLE_0:def 4, A3;
end;

definition
  let Q;
  func Cent Q -> Subset of Q equals
  Comm Q /\ Nucl Q;
  coherence;
end;

definition
  let Q1,Q2 be multLoop;
  let f be Function of Q1,Q2;
  attr f is unity-preserving means :Def28a:
    f.(1.Q1) = 1.Q2;
  attr f is quasi-homomorphic means :Def28b:
    for x,y being Element of Q1 holds f.(x * y) = (f.x) * (f.y);
end;

definition
  let Q1,Q2 be multLoop;
  let f be Function of Q1,Q2;
  attr f is homomorphic means
  f is unity-preserving quasi-homomorphic;
end;

registration
  let Q1,Q2 be multLoop;
  cluster unity-preserving quasi-homomorphic -> homomorphic
          for Function of Q1,Q2;
  coherence;
  cluster homomorphic -> unity-preserving quasi-homomorphic
          for Function of Q1,Q2;
  coherence;
end;

registration
  let Q1,Q2 be multLoop;
  cluster [#]Q1 --> 1.Q2 -> homomorphic for Function of Q1,Q2;
  coherence
  proof
    let f be Function of Q1,Q2 such that
A1: f = [#]Q1 --> 1.Q2;
    thus f.(1.Q1) = 1.Q2 by A1;
    thus thesis by A1;
  end;
end;

registration
  let Q1,Q2 be multLoop;
  cluster homomorphic for Function of Q1,Q2; 
  existence
  proof
    reconsider f = [#]Q1 --> 1.Q2 as Function of Q1,Q2;
    take f;
    thus thesis;
  end;
end;

definition
  let Q,Q2;
  let f be homomorphic Function of Q,Q2;
  func Ker f -> Subset of Q means
  :Def29:
  x in it iff f.x = 1.Q2;
  existence
  proof
    set N = {x : f.x = 1.Q2};
    N c= the carrier of Q
    proof
      let x be object;
      assume x in N;
      then ex x1 be Element of Q st x = x1 & f.x1 = 1.Q2;
      hence thesis;
    end;
    then reconsider N as Subset of Q;
    take N;
    x in N implies f.x = 1.Q2
    proof
      assume x in N;
      then ex x1 be Element of Q st x = x1 & f.x1 = 1.Q2;
      hence thesis;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let X1,X2 be Subset of Q;
    assume A1: for x being Element of Q holds x in X1 iff f.x = 1.Q2;
    assume A2: for x being Element of Q holds x in X2 iff f.x = 1.Q2;
    now
      let x be Element of Q;
      x in X1 iff f.x = 1.Q2 by A1;
      hence x in X1 iff x in X2 by A2;
    end;
    hence thesis by SUBSET_1:3;
  end;
end;

theorem Th13:
  for f being homomorphic Function of Q1,Q2 holds
  for x,y being Element of Q1 holds f.(x \ y) = f.x \ f.y
proof
  let f be homomorphic Function of Q1,Q2;
  let x,y be Element of Q1;
  f.x * f.(x \ y) = f.(x * (x \ y)) by Def28b;
  hence thesis;
end;

theorem Th14:
  for f being homomorphic Function of Q1,Q2 holds
  for x,y being Element of Q1 holds f.(x / y) = f.x / f.y
proof
  let f be homomorphic Function of Q1,Q2;
  let x,y be Element of Q1;
  f.(x / y) * f.y = f.((x / y) * y) by Def28b;
  hence thesis;
end;

theorem Th15:
  for f being homomorphic Function of Q1,Q2 st
  (for y be Element of Q2 holds ex x being Element of Q1 st f.x = y)
  &
  (for x,y,z be Element of Q1 holds a_op(x,y,z) in Ker f)
  holds
  Q2 is associative
proof
  let f be homomorphic Function of Q1,Q2;
  assume that A1: for y be Element of Q2 holds ex x being Element of Q1 st
    f.x = y
  and A2: for x,y,z be Element of Q1 holds a_op(x,y,z) in Ker f;
  thus Q2 is associative
  proof
    let x,y,z be Element of Q2;
    consider x1 being Element of Q1 such that A3: f.x1 = x  by A1;
    consider y1 being Element of Q1 such that A4: f.y1 = y by A1;
    consider z1 being Element of Q1 such that A5: f.z1 = z by A1;
    A6: a_op(x1,y1,z1) in Ker f by A2;
    a_op(x,y,z)= (f.x1 * f.(y1 * z1)) \ ((f.x1 * f.y1) * f.z1)
        by Def28b,A3,A4,A5
      .= f.(x1 * (y1 * z1)) \ ((f.x1 * f.y1) * f.z1) by Def28b
      .= f.(x1 * (y1 * z1)) \ (f.(x1 * y1) * f.z1) by Def28b
      .= f.(x1 * (y1 * z1)) \ f.((x1 * y1) * z1) by Def28b
      .= f.((x1 * (y1 * z1)) \ ((x1 * y1) * z1)) by Th13
      .= 1.Q2 by A6, Def29;
    hence thesis by Th9;
  end;
end;

theorem Th16:
  for Q1 being
    satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_aK1 satisfying_aK2 satisfying_aK3 multLoop
  holds
  for Q2 be multLoop holds
  for f being homomorphic Function of Q1,Q2 st
  (for y be Element of Q2 holds ex x being Element of Q1 st f.x = y)
  &
  Nucl Q1 c= Ker f
  holds
  Q2 is commutative multGroup
proof
  let Q1 be satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_aK1 satisfying_aK2 satisfying_aK3 multLoop;
  let Q2 be multLoop;
  let f be homomorphic Function of Q1,Q2;
  assume that A1: for y be Element of Q2 holds ex x being Element of Q1 st
    f.x = y
  and A2: Nucl Q1 c= Ker f;
  A3: Q2 is commutative
  proof
    let x,y be Element of Q2;
    consider x1 being Element of Q1 such that A4: f.x1 = x by A1;
    consider y1 being Element of Q1 such that A5: f.y1 = y by A1;
    K_op(x,y) = 1.Q2
    proof
      A6: K_op(x1,y1) in Ker f
      proof
        A7: K_op(x1,y1) in Nucl Q1
        proof
          now
            let u,w be Element of Q1;
            a_op(K_op(x1,y1),u,w) = 1.Q1 by Def19;
            hence K_op(x1,y1) * (u * w) = (K_op(x1,y1) * u) * w by Th9;
          end;
          then A8: K_op(x1,y1) in Nucl_l Q1 by Def22;
          now
            let u,w be Element of Q1;
            a_op(u,K_op(x1,y1),w) = 1.Q1 by Def20;
            hence u * (K_op(x1,y1) * w) = (u * K_op(x1,y1)) * w by Th9;
          end;
          then A9: K_op(x1,y1) in Nucl_m Q1 by Def23;
          now
            let u,w be Element of Q1;
            a_op(u,w,K_op(x1,y1)) = 1.Q1 by Def21;
            hence u * (w * K_op(x1,y1)) = (u * w) * K_op(x1,y1) by Th9;
          end;
          then K_op(x1,y1) in Nucl_r Q1 by Def24;
          hence thesis by A8,A9,Th12;
        end;
        thus thesis by A7, A2;
      end;
      K_op(x,y) = f.(y1 * x1) \ (f.x1 * f.y1) by Def28b,A4,A5
      .= f.(y1 * x1) \ f.(x1 * y1) by Def28b
      .= f.((y1 * x1) \ (x1 * y1)) by Th13
      .= 1.Q2 by A6, Def29;
      hence thesis;
    end;
    hence thesis by Th10;
  end;
  now
    let x1,y1,z1 be Element of Q1;
    a_op(x1,y1,z1) in Nucl Q1
    proof
       now
        let u,w be Element of Q1;
        a_op(a_op(x1,y1,z1),u,w) = 1.Q1 by Def15;
        hence (a_op(x1,y1,z1) * u) * w = a_op(x1,y1,z1) * (u * w) by Th9;
      end;
      then A10: a_op(x1,y1,z1) in Nucl_l Q1 by Def22;
      now
        let u,w be Element of Q1;
        a_op(u,a_op(x1,y1,z1),w) = 1.Q1 by Def16;
        hence (u * a_op(x1,y1,z1)) * w = u * (a_op(x1,y1,z1) * w) by Th9;
      end;
      then A11: a_op(x1,y1,z1) in Nucl_m Q1 by Def23;
      now
        let u,w be Element of Q1;
        a_op(u,w,a_op(x1,y1,z1)) = 1.Q1 by Def17;
        hence (u * w) * a_op(x1,y1,z1) = u * (w * a_op(x1,y1,z1)) by Th9;
      end;
      then a_op(x1,y1,z1) in Nucl_r Q1 by Def24;
      hence thesis by A10,A11,Th12;
    end;
    hence a_op(x1,y1,z1) in Ker f by A2;
  end;
  hence thesis by A3,Th15,A1;
end;

theorem Th17:
  for Q1 being satisfying_aa1 satisfying_aa2 satisfying_aa3 satisfying_Ka
    multLoop holds
  for Q2 be multLoop holds
  for f being homomorphic Function of Q1,Q2 st
  (for y be Element of Q2 holds ex x being Element of Q1 st f.x = y)
  &
  Cent Q1 c= Ker f
  holds
  Q2 is multGroup
proof
  let Q1 be satisfying_aa1 satisfying_aa2 satisfying_aa3 satisfying_Ka
    multLoop;
  let Q2 be multLoop;
  let f be homomorphic Function of Q1,Q2;
  assume that A1: for y be Element of Q2 holds ex x being Element of Q1 st
    f.x = y
  and A2: Cent Q1 c= Ker f;
  now
    let x1,y1,z1 be Element of Q1;
    a_op(x1,y1,z1) in Cent Q1
    proof
      now
        let u be Element of Q1;
        K_op(a_op(x1,y1,z1),u) = 1.Q1 by Def18;
        hence a_op(x1,y1,z1) * u = u * a_op(x1,y1,z1) by Th10;
      end;
      then A3: a_op(x1,y1,z1) in Comm Q1 by Def25;
      now
        let u,w be Element of Q1;
        a_op(a_op(x1,y1,z1),u,w) = 1.Q1 by Def15;
        hence (a_op(x1,y1,z1) * u) * w = a_op(x1,y1,z1) * (u * w) by Th9;
      end;
      then A4: a_op(x1,y1,z1) in Nucl_l Q1 by Def22;
      now
        let u,w be Element of Q1;
        a_op(u,a_op(x1,y1,z1),w) = 1.Q1 by Def16;
        hence (u * a_op(x1,y1,z1)) * w = u * (a_op(x1,y1,z1) * w) by Th9;
      end;
      then A5: a_op(x1,y1,z1) in Nucl_m Q1 by Def23;
      now
        let u,w be Element of Q1;
        a_op(u,w,a_op(x1,y1,z1)) = 1.Q1 by Def17;
        hence (u * w) * a_op(x1,y1,z1) = u * (w * a_op(x1,y1,z1)) by Th9;
      end;
      then a_op(x1,y1,z1) in Nucl_r Q1 by Def24;
      then a_op(x1,y1,z1) in Nucl Q1 by A4,A5,Th12;
      hence thesis by A3,XBOOLE_0:def 4;
    end;
    hence a_op(x1,y1,z1) in Ker f by A2;
  end;
  hence thesis by Th15,A1;
end;

:: following GROUP_2
definition
  let Q be non empty multLoopStr;
  mode SubLoopStr of Q -> non empty multLoopStr means :Def30:
  the carrier of it c= the carrier of Q &
  the multF of it = (the multF of Q)||the carrier of it &
  the OneF of it = the OneF of Q;
  existence
  proof
    take Q;
    thus thesis;
  end;
end;

registration
  let Q be multLoop;
  cluster well-unital invertible cancelable non empty strict
          for SubLoopStr of Q;
  existence
  proof
    reconsider Q1=the multLoopStr of Q as non empty multLoopStr;
    the multF of Q1 = (the multF of Q)||the carrier of Q1;
    then reconsider Q1 as SubLoopStr of Q by Def30;
    take Q1;
    now
      let x be Element of Q1;
      reconsider x1=x as Element of Q;
      x*1.Q1 = x1*1.Q & 1.Q1*x = 1.Q*x1;
      hence x * 1. Q1 = x & 1.Q1 * x = x;
    end;
    hence Q1 is well-unital;
    thus Q1 is invertible
      proof
         hereby let x,y be Element of Q1;
           reconsider x1=x,y1=y as Element of Q;
           reconsider z=x1 \ y1 as Element of Q1;
           take z;
           thus x*z = x1 * (x1 \ y1) .= y;
         end;
         hereby let x,y be Element of Q1;
           reconsider x1=x,y1=y as Element of Q;
           reconsider z=y1 / x1 as Element of Q1;
           take z;
           thus z*x = (y1 / x1) * x1 .= y;
         end;
      end;
    thus Q1 is cancelable
      proof
       thus Q1 is left_mult-cancelable
         proof
           let x be Element of Q1;
           let y,z be Element of Q1;
           reconsider x1=x,y1=y,z1=z as Element of Q;
           assume x*y=x*z;
           then x1*y1 = x1*z1;
           hence y=z by ALGSTR_0:def 20;
         end;
       thus Q1 is right_mult-cancelable
        proof
           let x be Element of Q1;
           let y,z be Element of Q1;
           reconsider x1=x,y1=y,z1=z as Element of Q;
           assume y*x=z*x;
           then y1*x1 = z1*x1;
           hence y=z by ALGSTR_0:def 21;
         end;
     end;
    thus thesis;
  end;
end;

definition
  let Q be multLoop;
  mode SubLoop of Q is well-unital invertible cancelable SubLoopStr of Q;
end;

definition
  let Q be non empty multLoopStr;
  let H be SubLoopStr of Q;
  let A be Subset of H;
  func @ A -> Subset of Q equals A;
  coherence
  proof
    the carrier of H c= the carrier of Q by Def30;
    hence thesis by XBOOLE_1:1;
  end;
end;

defpred RQ[multLoop,Subset of $1,object] means
  ex y,z be Element of $1 st y in $2 & z in $2 &
    ($3 = y * z or $3 = y \z or $3 = y / z);

definition
  let Q;
  let H1,H2 be Subset of Q;
  func loopclose1(H1,H2) -> Subset of Q means
  :Def32:
  x in it iff
  x in H1 or x = 1.Q or
  ex y,z st y in H2 & z in H2 & (x = y * z or x = y \ z or x = y / z);
  existence
  proof
    set H3 = {x : x in H1 or x = 1.Q or RQ[Q,H2,x]};
    H3 c= the carrier of Q
    proof
      let x be object;
      assume x in H3;
      then ex x1 being Element of Q st
       x = x1 & (x1 in H1 or x1 = 1.Q or RQ[Q,H2,x1]);
      hence thesis;
    end;
    then reconsider H3 as Subset of Q;
    take H3;
    let x be Element of Q;
    thus x in H3 implies x in H1 or x = 1.Q or RQ[Q,H2,x]
    proof
      assume x in H3;
      then ex x1 being Element of Q st
       x = x1 & (x1 in H1 or x1 = 1.Q or RQ[Q,H2,x1]);
      hence thesis;
    end;
    thus thesis;
  end;
  uniqueness
  proof
    let H3,H4 be Subset of Q;
    assume that
    A1: x in H3 iff x in H1 or x = 1.Q or RQ[Q,H2,x]
    and
    A2: x in H4 iff x in H1 or x = 1.Q or RQ[Q,H2,x];
    now
      let x be Element of Q;
      x in H3 iff x in H1 or x = 1.Q or RQ[Q,H2,x] by A1;
      hence x in H3 iff x in H4 by A2;
    end;
    hence thesis by SUBSET_1:3;
  end;
end;

definition
  let Q;
  let H be Subset of Q;
  func lp H -> strict SubLoop of Q means
  :Def33:
  H c= [#]it &
  for H2 be SubLoop of Q st H c= [#]H2 holds [#]it c= [#]H2;
  existence
  proof
    deffunc F(Subset of Q) = loopclose1(H,$1);
    consider f be Function of bool the carrier of Q,bool the carrier of Q
    such that
    A1: for X being Subset of Q holds f.X = F(X)
    from FUNCT_2:sch 4;
    f is c=-monotone
    proof
      let a1, b1 be set such that
      A2: a1 in dom f & b1 in dom f & a1 c= b1;
      thus f.a1 c= f.b1
      proof
        reconsider a2 = a1,b2=b1  as Subset of Q by A2, FUNCT_2:def 1;
        let x be object;
        assume x in f.a1;
        then x in F(a2) by A1;
        then x in H or x = 1.Q or RQ[Q,a2,x] by Def32;
        then x in F(b2) by A2,Def32;
        hence x in f.b1 by A1;
      end;
    end;
    then reconsider f as
    c=-monotone Function of bool the carrier of Q,bool the carrier of Q;
    set LFP= lfp(the carrier of Q,f);
     LFP is_a_fixpoint_of f by KNASTER:4;
    then A3: LFP in dom f & LFP = f.(LFP) & f.(LFP) = F(LFP)
      by ABIAN:def 3,A1;
    A4: 1.Q in F(LFP) by Def32;
    reconsider ONE=1.Q as Element of LFP by A3,Def32;
    set mm = (the multF of Q)||LFP;
    now let x be set such that A5: x in [: LFP,LFP:];
      consider x1,x2 be object such that
      A6:x1 in LFP & x2 in LFP & x=[x1,x2] by A5,ZFMISC_1:def 2;
      reconsider x1,x2 as Element of Q by A6;
      x1*x2 in F(LFP) by A6,Def32;
      hence (the multF of Q).x in LFP by A6,A3;
    end;
    then
    LFP is Preserv of the multF of Q by REALSET1:def 1;
    then reconsider mm as BinOp of LFP by REALSET1:2;
    set LP =  multLoopStr(#LFP,mm,ONE#);
    reconsider LP as non empty SubLoopStr of Q by A4,A3,Def30;
    LP is SubLoop of Q
    proof
      now
        let x be Element of LP;
        x in the carrier of LP;
        then reconsider x1=x as Element of Q;
        x*1.LP = x1*1.Q & 1.LP*x = 1.Q*x1 by RING_3:1;
        hence x * 1. LP = x & 1.LP * x = x;
      end;
      then A7: LP is well-unital;
      A8: LP is invertible
      proof
        hereby let x,y be Element of LP;
          x in the carrier of LP &   y in the carrier of LP;
          then reconsider x1=x,y1=y as Element of Q;
          reconsider z=x1 \ y1 as Element of LP by Def32,A3;
          take z;
          thus x*z = x1 * (x1 \ y1) by RING_3:1
          .= y;
        end;
        hereby let x,y be Element of LP;
          x in the carrier of LP &   y in the carrier of LP;
          then reconsider x1=x,y1=y as Element of Q;
         reconsider z=y1 / x1 as Element of LP by Def32,A3;
          take z;
          thus z*x = (y1 / x1) * x1 by RING_3:1
          .= y;
        end;
      end;
      LP is cancelable
      proof
        thus LP is left_mult-cancelable
        proof
          let x be Element of LP;
          let y,z be Element of LP;
          x in the carrier of LP & y in the carrier of LP &
            z in the carrier of LP;
          then reconsider x1=x,y1=y,z1=z as Element of Q;
          x1*y1 = x*y & x1*z1 = x*z by RING_3:1;
          hence thesis by ALGSTR_0:def 20;
        end;
        let x be Element of LP;
        let y,z be Element of LP;
        x in the carrier of LP & y in the carrier of LP &
          z in the carrier of LP;
        then reconsider x1=x,y1=y,z1=z as Element of Q;
        y1*x1 = y*x & z1*x1 = z*x by RING_3:1;
        hence thesis by ALGSTR_0:def 21;
      end;
      hence thesis by A7,A8;
    end;
    then
    reconsider LP as strict SubLoop of Q;
    take LP;
    thus H c= [#]LP by A3,Def32 ;
    let H2 be SubLoop of Q such that
    A9: H c= [#]H2;
    reconsider H2c = [#]H2 as Subset of Q by Def30;
    f.([#]H2) c= [#]H2
    proof
      let x be object;
      assume x in f.([#]H2);
      then A10:x in F(H2c) by A1;
      then reconsider xx=x as Element of Q;
      per cases by A10,Def32;
      suppose x in H;
        hence thesis by A9;
      end;
      suppose x = 1.Q;
        then x = 1.H2 by Def30;
        hence thesis;
      end;
      suppose RQ[Q,H2c,x];
        then consider y,z such that
        A11: y in H2c & z in H2c &
        (x = y * z or x = y \ z or x = y / z);
        reconsider y1=y,z1=z as Element of H2 by A11;
        y1\z1 in H2c & y1/z1 in H2c;
        then reconsider yz =y1\z1,YZ =y1/z1 as Element of Q;
       the multF of H2 = (the multF of Q)||H2c by Def30;
       then y*z = y1*z1 & y *yz = y1 * (y1\z1)= z &  YZ * z=  (y1/z1)* z1= y
       by RING_3:1;
       hence thesis by A11;
      end;
    end;
    then f.(H2c) c= H2c;
    hence [#]LP c= [#]H2 by KNASTER:6;
  end;
  uniqueness
  proof
    let IT1,IT2 be strict SubLoop of Q such that
    A12: H c= [#]IT1 &
    for H2 be SubLoop of Q st H c= [#]H2 holds [#]IT1 c= [#]H2
    and A13: H c= [#]IT2 &
    for H2 be SubLoop of Q st H c= [#]H2 holds [#]IT2 c= [#]H2;
    A14: [#]IT1 = [#]IT2 by A12,A13;
    A15: the OneF of IT1 = 1.Q by Def30
     .= the OneF of IT2 by Def30;
    the multF of IT1 = (the multF of Q)||(the carrier of IT1) by Def30
    .= the multF of IT2 by Def30,A14;
    hence thesis by A14,A15;
  end;
end;

theorem Th18:
  for H being Subset of Q st
  1.Q in H & (for x,y st x in H & y in H holds x * y in H)
  & (for x,y st x in H & y in H holds x \ y in H)
  & (for x,y st x in H & y in H holds x / y in H)
  holds
  [#]lp H = H
proof
  let H be Subset of Q;
  assume that
  A1: 1.Q in H
  and
  A2: for x,y st x in H & y in H holds x * y in H
  and
  A3: for x,y st x in H & y in H holds x \ y in H
  and
  A4: for x,y st x in H & y in H holds x / y in H;
  reconsider ONE=1.Q as Element of H by A1;
  set mm = (the multF of Q)||H;
  now let x be set such that A5: x in [: H,H:];
    consider x1,x2 be object such that
    A6:x1 in H & x2 in H & x=[x1,x2] by A5,ZFMISC_1:def 2;
    reconsider x1,x2 as Element of Q by A6;
    x1*x2 in H by A6,A2;
    hence (the multF of Q).x in H by A6;
  end;
  then
  H is Preserv of the multF of Q by REALSET1:def 1;
  then reconsider mm as BinOp of H by REALSET1:2;
  set LP = multLoopStr(#H,mm,ONE#);
  reconsider LP as non empty SubLoopStr of Q by A1,Def30;
  LP is SubLoop of Q
  proof
    now
      let x be Element of LP;
      x in the carrier of LP;
      then reconsider x1=x as Element of Q;
      x*1.LP = x1*1.Q & 1.LP*x = 1.Q*x1 by RING_3:1;
      hence x * 1. LP = x & 1.LP * x = x;
    end;
    then  A7: LP is well-unital;
    A8: LP is invertible
    proof
      hereby let x,y be Element of LP;
        x in the carrier of LP &   y in the carrier of LP;
        then reconsider x1=x,y1=y as Element of Q;
        reconsider z=x1 \ y1 as Element of LP by A3;
        take z;
        thus x*z = x1 * (x1 \ y1) by RING_3:1
        .= y;
      end;
      hereby let x,y be Element of LP;
        x in the carrier of LP &   y in the carrier of LP;
        then reconsider x1=x,y1=y as Element of Q;
        reconsider z=y1 / x1 as Element of LP by A4;
        take z;
        thus z*x = (y1 / x1) * x1 by RING_3:1
        .= y;
      end;
    end;
    LP is  cancelable
    proof
      thus LP is left_mult-cancelable
      proof
        let x be Element of LP;
        let y,z be Element of LP;
        x in the carrier of LP
        & y in the carrier of LP
        & z in the carrier of LP;
        then reconsider x1=x,y1=y,z1=z as Element of Q;
        x1*y1 = x*y & x1*z1 = x*z by RING_3:1;
        hence thesis by ALGSTR_0:def 20;
      end;
      let x be Element of LP;
      let y,z be Element of LP;
      x in the carrier of LP & y in the carrier of LP & z in the carrier of LP;
      then reconsider x1=x,y1=y,z1=z as Element of Q;
      y1*x1 = y*x & z1*x1 = z*x by RING_3:1;
      hence thesis by ALGSTR_0:def 21;
    end;
    hence thesis by A7,A8;
  end;
  then reconsider LP as strict SubLoop of Q;
  [#](lp H) c= [#]LP = H by Def33;
  hence thesis by Def33;
end;

theorem Th19:
  for f being homomorphic Function of Q,Q2 holds
  [#]lp (Ker f) = Ker f
  proof
    let f be homomorphic Function of Q,Q2;
    f.(1.Q) = 1.Q2 by Def28a;
    then A1: 1.Q in Ker f by Def29;
    A2: for x,y st x in Ker f & y in Ker f holds x * y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A3: x in Ker f
      and
      A4: y in Ker f;
      f.(x * y) = f.x * f.y by Def28b
      .= 1.Q2 * f.y by Def29,A3
      .= 1.Q2 by Def29,A4;
      hence x*y in Ker f by Def29;
    end;
    A5: for x,y st x in Ker f & y in Ker f holds x \ y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A6: x in Ker f
      and
      A7: y in Ker f;
      f.(x \ y) = f.x \ f.y by Th13
      .= 1.Q2 \ f.y by Def29,A6
      .= 1.Q2 by Def29,A7;
      hence x\y in Ker f by Def29;
    end;
    for x,y st x in Ker f & y in Ker f holds x / y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A8: x in Ker f
      and
      A9: y in Ker f;
      f.(x / y) = f.x / f.y by Th14
      .= f.x / 1.Q2 by Def29,A9
      .= 1.Q2 by Def29,A8;
      hence x/y in Ker f by Def29;
    end;
    hence thesis by Th18,A1,A2,A5;
  end;

theorem Th20a:
  1.Q in Nucl_l Q
proof
  for y,z holds (1.Q * y) * z = 1.Q * (y * z);
  hence thesis by Def22;
end;

theorem Th20b:
  1.Q in Nucl_m Q
proof
  for x,z holds (x * 1.Q) * z = x * (1.Q * z);
  hence thesis by Def23;
end;

theorem Th20c:
  1.Q in Nucl_r Q
proof
  for x,y holds (x * y) * 1.Q = x * (y * 1.Q);
  hence thesis by Def24;
end;

theorem Th20:
  1.Q in Nucl Q
proof
  A1: 1.Q in Nucl_l Q by Th20a;
  1.Q in Nucl_m Q by Th20b;
  hence thesis by A1,Th12,Th20c;
end;

registration
  let Q;
  cluster Nucl_l Q -> non empty;
  coherence by Th20a;
  cluster Nucl_m Q -> non empty;
  coherence by Th20b;
  cluster Nucl_r Q -> non empty;
  coherence by Th20c;
  cluster Nucl Q -> non empty;
  coherence by Th20;
end;

theorem Th21:
  x in Nucl Q & y in Nucl Q implies x * y in Nucl Q
proof
  assume that
  A1: x in Nucl Q
  and
  A2: y in Nucl Q;
  A3: x in Nucl_l Q by Th12,A1;
  A4: x in Nucl_m Q by Th12,A1;
  A5: x in Nucl_r Q by Th12,A1;
  A6: y in Nucl_l Q by Th12,A2;
  A7: y in Nucl_m Q by Th12,A2;
  A8: y in Nucl_r Q by Th12,A2;
  for z,w holds ((x * y) * z) * w = (x * y) * (z * w)
  proof
    let z,w;
    ((x * y) * z) * w = (x * (y * z)) * w by A3,Def22
    .= x * ((y * z) * w) by A3,Def22
    .= x * (y * (z * w)) by A6,Def22
    .= (x * y) * (z * w) by A3,Def22;
    hence thesis;
  end;
  then A9: x * y in Nucl_l Q by Def22;
  for z,w holds (z * (x * y)) * w = z * ((x * y) * w)
  proof
    let z,w;
    (z * (x * y)) * w = ((z * x) * y) * w by A4,Def23
    .= (z * x) * (y * w) by A7,Def23
    .= z * (x * (y * w)) by A4,Def23
    .= z * ((x * y) * w) by A7,Def23;
    hence thesis;
  end;
  then A10: x * y in Nucl_m Q by Def23;
  for z,w holds (z * w) * (x * y) = z * (w * (x * y))
  proof
    let z,w;
    (z * w) * (x * y) = ((z * w) * x) * y by A8,Def24
    .= (z * (w * x)) * y by A5,Def24
    .= z * ((w * x) * y) by A8,Def24
    .= z * (w * (x * y)) by A8,Def24;
    hence thesis;
  end;
  then x * y in Nucl_r Q by Def24;
  hence thesis by Th12,A9,A10;
end;

theorem Th22:
  x in Nucl Q & y in Nucl Q implies x \ y in Nucl Q
proof
  assume that
  A1: x in Nucl Q
  and
  A2: y in Nucl Q;
  A3: x in Nucl_l Q by Th12,A1;
  A4: x in Nucl_m Q by Th12,A1;
  A5: x in Nucl_r Q by Th12,A1;
  A6: y in Nucl_l Q by Th12,A2;
  A7: y in Nucl_m Q by Th12,A2;
  A8: y in Nucl_r Q by Th12,A2;
  for z,w holds ((x \ y) * z) * w = (x \ y) * (z * w)
  proof
    let z,w;
    x * (((x \ y) * z) * w) = (x * ((x \ y) * z)) * w by A3,Def22
    .= ((x * (x \ y)) * z) * w by A3,Def22
    .= (x * (x \ y)) * (z * w) by A6,Def22
    .= x * ((x \ y) * (z * w)) by A3,Def22;
    hence thesis by Th1;
  end;
  then A9: x \ y in Nucl_l Q by Def22;
  for z,w holds (z * (x \ y)) * w = z * ((x \ y) * w)
  proof
    let z,w;
    (z * (x \ y)) * w = (((z / x) * x) * (x \ y)) * w
    .= ((z / x) * (x * (x \ y))) * w by A4,Def23
    .= (z / x) * ((x * (x \ y)) * w) by A7,Def23
    .= (z / x) * (x * ((x \ y) * w)) by A3,Def22
    .= ((z / x) * x) * ((x \ y) * w) by A4,Def23
    .= z * ((x \ y) * w);
    hence thesis;
  end;
  then A10: x \ y in Nucl_m Q by Def23;
  for z,w holds (z * w) * (x \ y) = z * (w * (x \ y))
  proof
    let z,w;
    (z * w) * (x \ y) = (z * ((w / x) * x)) * (x \ y)
    .= ((z * (w / x)) * x) * (x \ y) by A5,Def24
    .= (z * (w / x)) * (x * (x \ y)) by A4,Def23
    .= z * ((w / x) * (x * (x \ y)))by A8,Def24
    .= z * (((w / x) * x) * (x \ y)) by A4,Def23
    .= z * (w * (x \ y));
    hence thesis;
  end;
  then x \ y in Nucl_r Q by Def24;
  hence thesis by Th12,A9,A10;
end;

theorem Th23:
  x in Nucl Q & y in Nucl Q implies x / y in Nucl Q
proof
  assume that
  A1: x in Nucl Q
  and
  A2: y in Nucl Q;
  A3: x in Nucl_l Q by Th12,A1;
  A4: x in Nucl_m Q by Th12,A1;
  A5: x in Nucl_r Q by Th12,A1;
  A6: y in Nucl_l Q by Th12,A2;
  A7: y in Nucl_m Q by Th12,A2;
  A8: y in Nucl_r Q by Th12,A2;
  for z,w holds ((x / y) * z) * w = (x / y) * (z * w)
  proof
    let z,w;
    ((x / y) * z) * w = ((x / y) * (y * (y \ z))) * w
    .= (((x / y) * y) * (y \ z)) * w by A7,Def23
    .= ((x / y) * y) * ((y \ z) * w) by A3,Def22
    .= (x / y) * (y * ((y \ z) * w)) by A7,Def23
    .= (x / y) * ((y * (y \ z)) * w) by A6,Def22
    .= (x / y) * (z * w);
    hence thesis;
  end;
  then A9: x / y in Nucl_l Q by Def22;
  for z,w holds (z * (x / y)) * w = z * ((x / y) * w)
  proof
    let z,w;
    (z * (x / y)) * w = (z * (x / y)) * (y * (y \ w))
    .= ((z * (x / y)) * y) * (y \ w) by A7,Def23
    .= (z * ((x / y) * y)) * (y \ w) by A8,Def24
    .= z * (((x / y) * y) * (y \ w)) by A4, Def23
    .= z * ((x / y) * (y * (y \ w))) by A7,Def23
    .= z * ((x / y) * w);
    hence thesis;
  end;
  then A10: x / y in Nucl_m Q by Def23;
  for z,w holds (z * w) * (x / y) = z * (w * (x / y))
  proof
    let z,w;
    ((z * w) * (x / y)) * y = (z * w) * ((x / y) * y) by A8,Def24
    .= z * (w * ((x / y) * y)) by A5,Def24
    .= z * ((w * (x / y)) * y) by A8,Def24
    .= (z * (w * (x / y))) * y by A8,Def24;
    hence thesis by Th2;
  end;
  then x / y in Nucl_r Q by Def24;
  hence thesis by Th12,A9,A10;
end;

theorem Th24:
  [#]lp (Nucl Q) = Nucl Q
proof
  A1: 1.Q in Nucl Q by Th20;
  A2: for x,y st x in Nucl Q & y in Nucl Q holds x * y in Nucl Q
    by Th21;
  A3: for x,y st x in Nucl Q & y in Nucl Q holds x \ y in Nucl Q
    by Th22;
   for x,y st x in Nucl Q & y in Nucl Q holds x / y in Nucl Q
     by Th23;
  hence thesis by Th18,A1,A2,A3;
end;

theorem Th25:
  [#]lp (Cent Q) = Cent Q
proof
  A1: 1.Q in Cent Q
  proof
    A2: 1.Q in Nucl Q by Th20;
    for y holds 1.Q * y = y * 1.Q;
    then 1.Q in Comm Q by Def25;
    hence thesis by XBOOLE_0:def 4, A2;
  end;
  A3: for x,y st x in Cent Q & y in Cent Q holds x * y in Cent Q
  proof
    let x,y;
    assume that
    A4: x in Cent Q
    and
    A5: y in Cent Q;
    A6: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A4;
    A7: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A5;
    A8: x in Nucl_l Q by Th12,A6;
    A9: y in Nucl_m Q & y in Nucl_r Q by Th12,A7;
    for z holds (x * y) * z = z * (x * y)
    proof
      let z;
      (x * y) * z = x * (y * z) by A9,Def23
      .= x * (z * y) by A7,Def25
      .= (x * z) * y by A8,Def22
      .= (z * x) * y by A6,Def25
      .= z * (x * y) by A9,Def24;
      hence thesis;
    end;
    then A10: x * y in Comm Q by Def25;
    x * y in Nucl Q by Th21,A6,A7;
    hence x * y in Cent Q by XBOOLE_0:def 4,A10;
  end;
  A11: for x,y st x in Cent Q & y in Cent Q holds x \ y in Cent Q
  proof
    let x,y;
    assume that
    A12: x in Cent Q
    and
    A13: y in Cent Q;
    A14: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A12;
    A15: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A13;
    A16: x in Nucl_m Q by Th12,A14;
    for z holds (x \ y) * z = z * (x \ y)
    proof
      let z;
      (x \ y) * z = (x \ y) * ((z / x) * x)
      .= (x \ y) * (x * (z / x)) by A14,Def25
      .= ((x \ y) * x) * (z / x) by A16,Def23
      .= (x * (x \ y)) * (z / x) by A14,Def25
      .= (z / x) * (x * (x \ y)) by A15,Def25
      .= ((z / x) * x) * (x \ y) by A16,Def23
      .= z * (x \ y);
      hence thesis;
    end;
    then A17: x \ y in Comm Q by Def25;
    x \ y in Nucl Q by Th22,A14,A15;
    hence x \ y in Cent Q by XBOOLE_0:def 4, A17;
  end;
  for x,y st x in Cent Q & y in Cent Q holds x / y in Cent Q
  proof
    let x,y;
    assume that
    A18: x in Cent Q
    and
    A19: y in Cent Q;
    A20: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A18;
    A21: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A19;
    A22: y in Nucl_m Q by Th12,A21;
    for z holds (x / y) * z = z * (x / y)
    proof
      let z;
      thus (x / y) * z = (x / y) * ((z / y) * y)
      .= (x / y) * (y * (z / y)) by A21,Def25
      .= ((x / y) * y) * (z / y) by A22,Def23
      .= (z / y) * ((x / y) * y) by A20,Def25
      .= (z / y) * (y * (x / y)) by A21,Def25
      .= ((z / y) * y) * (x / y) by A22,Def23
      .= z * (x / y);
    end;
    then A23: x / y in Comm Q by Def25;
    x / y in Nucl Q by Th23,A20,A21;
    hence x / y in Cent Q by XBOOLE_0:def 4,A23;
  end;
  hence thesis by Th18,A1,A3,A11;
end;

begin :: Multiplicative Mappings and Cosets

::We now define a set Mlt Q of multiplicative mappings of Q
::and cosets (mostly following Albert 1943 for cosets).

definition
  let X be functional set;
  attr X is composition-closed means
  :Def34:
  for f,g being Element of X st f in X & g in X holds f*g in X;
  attr X is inverse-closed means
  :Def35:
  for f being Element of X st f in X holds f" in X;
end;

registration
  let A be set;
  cluster {id A} -> composition-closed inverse-closed;
  coherence
  proof
    set I = id A;
    thus {I} is composition-closed
    proof
      let f,g be Element of {I};
      f = I & g = I by TARSKI:def 1;
      hence thesis by SYSREL:12;
    end;
    let f be Element of {I};
    f = I by TARSKI:def 1;
    then f is Permutation of A & I*f = I by SYSREL:12;
    then f" = I by FUNCT_2:60;
    hence thesis by TARSKI:def 1;
  end;
end;

registration
  cluster composition-closed inverse-closed non empty for functional set;
  existence
  proof
    take {id the set};
    thus thesis;
  end;
end;

registration
  let Q be multLoopStr;
  cluster composition-closed inverse-closed non empty for Subset of Funcs(Q,Q);
  existence
  proof
    set I = id Q;
    I in Funcs(Q,Q) by FUNCT_2:126;
    then reconsider X = {I} as Subset of Funcs(Q,Q) by SUBSET_1:33;
    take X;
    thus thesis;
  end;
end;

definition
  let Q be non empty multLoopStr;
  let H be Subset of Q;
  let S be Subset of Funcs(Q,Q);
  pred H left-right-mult-closed S means
  for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in S
  &
    (curry' (the multF of Q)).u in S;
end;

defpred PQ[multLoopStr,Subset of $1,
          Subset of Funcs($1,$1),object]
   means
    (ex u be Element of $1 st u in $2 & $4 = (curry' (the multF of $1)).u)
    or
    (ex u be Element of $1 st u in $2 & $4 = (curry (the multF of $1)).u)
    or
    (ex g,h be Permutation of $1 st g in $3 & h in $3 & $4 = g*h)
    or
    (ex g be Permutation of $1 st g in $3 & $4 = g");

definition
  let Q be non empty multLoopStr;
  let H be Subset of Q;
  let S be Subset of Funcs(Q,Q);
  func MltClos1(H,S) -> Subset of Funcs(Q,Q) means :Def37:
  for f being object holds
  f in it iff
    (ex u be Element of Q st u in H & f = (curry' (the multF of Q)).u)
  or
    (ex u be Element of Q st u in H & f = (curry (the multF of Q)).u)
  or
    (ex g,h be Permutation of Q st g in S & h in S & f = g*h)
  or
    (ex g be Permutation of Q st g in S & f = g");
  existence
  proof
    set mQ = the multF of Q;
    set LH = {(curry' mQ).u where u is Element of Q : u in H};
    set RH = {(curry mQ).u where u is Element of Q : u in H};
    set SC = {g*h where g,h is Permutation of Q : g in S & h in S};
    set SI = {g" where g is Permutation of Q : g in S};
    set Y = LH \/ RH \/ SC \/ SI;
    A1: LH c=  Funcs(Q,Q)
    proof
      let f be object;
      assume f in LH;
      then ex u being Element of Q st f = (curry' mQ).u & u in H;
      hence thesis;
    end;
    RH c= Funcs(Q,Q)
    proof
      let f be object;
      assume f in RH;
      then ex u being Element of Q st f = (curry mQ).u & u in H;
      hence thesis;
    end;
    then A2: LH \/ RH is Subset of Funcs(Q,Q) by A1, XBOOLE_1:8;
    SC c= Funcs(Q,Q)
    proof
      let f be object;
      assume f in SC;
      then ex g,h being Permutation of Q st f = g*h & g in S & h in S;
      hence thesis by FUNCT_2:9;
    end;
    then A3: LH \/ RH \/ SC is Subset of Funcs(Q,Q) by A2, XBOOLE_1:8;
    SI c= Funcs(Q,Q)
    proof
      let f be object;
      assume f in SI;
      then ex g being Permutation of Q st f = g" & g in S;
      hence thesis by FUNCT_2:9;
    end;
    then reconsider Y as Subset of Funcs(Q,Q) by A3, XBOOLE_1:8;
    take Y;
    let f be object;
    thus f in Y implies PQ[Q,H,S,f]
    proof
      assume f in Y;
      then f in LH \/ RH \/ SC or f in SI by XBOOLE_0:def 3;
      then f in LH \/ RH or f in SC or f in SI by XBOOLE_0:def 3;
      then per cases by XBOOLE_0:def 3;
      suppose f in LH;
        then ex u being Element of Q st f = (curry' mQ).u & u in H;
        hence thesis;
      end;
      suppose f in RH;
        then ex u being Element of Q st f = (curry mQ).u & u in H;
        hence thesis;
      end;
      suppose f in SC;
        then ex g,h being Permutation of Q st f = g*h & g in S & h in S;
        hence thesis;
      end;
      suppose f in SI;
        then ex g being Permutation of Q st f = g" & g in S;
        hence thesis;
      end;
    end;
    assume PQ[Q,H,S,f];
    then f in LH or f in RH or f in SC or f in SI;
    then f in LH\/RH or f in SC or f in SI by XBOOLE_0:def 3;
    then f in LH\/RH\/SC or f in SI by XBOOLE_0:def 3;
    hence  f in Y by XBOOLE_0:def 3;
  end;
  uniqueness
  proof
    let S1,S2 be Subset of Funcs(Q,Q);
    assume that
    A4: for f being object holds f in S1 iff PQ[Q,H,S,f]
    and
    A5: for f being object holds f in S2 iff PQ[Q,H,S,f];
    now
      let f be Element of Funcs(Q,Q);
      f in S1 iff PQ[Q,H,S,f] by A4;
      hence f in S1 iff f in S2 by A5;
    end;
    hence thesis by SUBSET_1:3;
  end;
end;

theorem Th26:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  st
  for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  phi is c=-monotone
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  let a1, b1 be set such that
  A2: a1 in dom phi & b1 in dom phi & a1 c= b1;
  thus phi.a1 c= phi.b1
  proof
    reconsider a2 = a1, b2=b1 as Subset of Funcs(Q,Q)
        by A2,FUNCT_2:def 1;
    let f be object;
    assume f in phi.a1;
    then f in MltClos1(H,a2) by A1;
    then PQ[Q,H,a2,f]  by Def37;
    then f in MltClos1(H,b2) by A2,Def37;
    hence thesis by A1;
  end;
end;

theorem Th27:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q) st
    for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  for Y being Subset of Funcs(Q,Q) st phi.(Y) c= Y
  holds
  (for u being Element of Q st u in H holds (curry (the multF of Q)).u in Y)
  &
  (for u being Element of Q st u in H holds (curry' (the multF of Q)).u in Y)
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  let Y be Subset of Funcs(Q,Q);
  assume phi.(Y) c= Y;
  then A2: MltClos1(H,Y) c= Y by A1;
  thus for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in Y
  proof
    let u be Element of Q;
    assume u in H;
    then (curry (the multF of Q)).u in MltClos1(H,Y) by Def37;
    hence thesis by A2;
  end;
    let u be Element of Q;
    assume u in H;
    then (curry' (the multF of Q)).u in MltClos1(H,Y) by Def37;
    hence thesis by A2;
end;

theorem Th28:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  st
  for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  for Y being Subset of Funcs(Q,Q)
  st
  for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S
  holds
  for f being Element of Funcs(Q,Q) st
  f in Y holds
  f is Permutation of Q
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  let Y be Subset of Funcs(Q,Q);
  assume A2: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S;
  set SP =  the set of all f where f is Permutation of Q;
  SP c= Funcs(Q,Q)
  proof
    let f be object;
    assume f in SP;
    then consider g being Permutation of Q such that
    A3: f = g & not contradiction;
    thus thesis by FUNCT_2:9,A3;
  end;
  then reconsider SP as Subset of Funcs(Q,Q);
  phi.(SP) c= SP
  proof
    let f be object;
    assume f in phi.(SP);
    then f in MltClos1(H,SP) by A1;
    then per cases by Def37;
    suppose ex u be Element of Q st u in H & f = (curry' (the multF of Q)).u;
      then consider u being Element of Q such that
      A4: u in H & f = (curry' (the multF of Q)).u;
      reconsider f as Function of Q,Q by A4;
      deffunc G(Element of Q) = $1 / u;
      consider g be Function of Q,Q such that
      A5: for x being Element of Q holds g.x = G(x) from FUNCT_2:sch 4;
      for x being Element of Q holds (g*f).x = (id (the carrier of Q)).x
      proof
        let x be Element of Q;
        (g * f).x = g.(f.x) by FUNCT_2:15
        .= g.(x * u) by FUNCT_5:70,A4
        .= G(x * u) by A5
        .= (id (the carrier of Q)).x;
        hence thesis;
      end;
      then A6: g * f = id (the carrier of Q) by FUNCT_2:def 8;
      for x being Element of Q holds (f*g).x = (id (the carrier of Q)).x
      proof
        let x be Element of Q;
        (f * g).x = f.(g.x) by FUNCT_2:15
        .= g.x * u by FUNCT_5:70,A4
        .= G(x) * u by A5
        .= (id (the carrier of Q)).x;
        hence thesis;
      end;
      then rng f = the carrier of Q by FUNCT_2:18,def 8;
      then f is Permutation of the carrier of Q by FUNCT_2:57,A6, FUNCT_2:31;
      hence thesis;
    end;
    suppose ex u be Element of Q st u in H & f = (curry (the multF of Q)).u;
      then consider u being Element of Q such that
      A7: u in H & f = (curry (the multF of Q)).u;
      reconsider f as Function of Q,Q by A7;
      deffunc G(Element of Q) = u \ $1;
      consider g be Function of Q,Q such that
      A8: for x being Element of Q holds g.x = G(x) from FUNCT_2:sch 4;
      A9: for x being Element of Q holds
        (g*f).x = (id (the carrier of Q)).x
      proof
        let x be Element of Q;
        (g * f).x = g.(f.x) by FUNCT_2:15
        .= g.(u * x) by FUNCT_5:69,A7
        .= G(u * x) by A8
        .= (id (the carrier of Q)).x;
        hence thesis;
      end;
      A10: for x being Element of Q holds
        (f*g).x = (id (the carrier of Q)).x
      proof
        let x be Element of Q;
        (f * g).x = f.(g.x) by FUNCT_2:15
        .= u * g.x by FUNCT_5:69,A7
        .= u * G(x) by A8
        .= (id (the carrier of Q)).x;
        hence thesis;
      end;
      A11: f is one-to-one by A9, FUNCT_2:31,FUNCT_2:def 8;
      rng f = the carrier of Q by A10, FUNCT_2:18,FUNCT_2:def 8;
      then f is Permutation of the carrier of Q by FUNCT_2:57,A11;
      hence thesis;
    end;
    suppose ex g,h be Permutation of the carrier of Q st g in SP &
        h in SP & f = g*h;
      hence thesis;
    end;
    suppose ex g be Permutation of the carrier of Q st g in SP & f = g";
      hence thesis;
    end;
  end;
  then A12: Y c= SP by A2;
  let f be Element of Funcs(Q,Q);
  assume f in Y;
  then f in SP by A12;
  then ex g being Permutation of Q st f = g;
  hence thesis;
end;

theorem Th29:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  st
  for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  for Y being Subset of Funcs(Q,Q)
  st
  Y is_a_fixpoint_of phi
  & for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S
  holds
  Y is composition-closed
  &
  Y is inverse-closed
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  let Y be Subset of Funcs(Q,Q);
  assume Y is_a_fixpoint_of phi;
  then A2: Y in dom phi & Y = phi.(Y) & phi.(Y) = MltClos1(H,Y)
    by ABIAN:def 3,A1;
  assume A3: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S;
  thus Y is composition-closed
  proof
    let f,g be Element of Y;
    assume A4: f in Y & g in Y;
    then f is Permutation of Q & g is Permutation of Q by Th28,A1,A3;
    hence f * g in Y by A2,Def37,A4;
  end;
    let f be Element of Y;
    assume A5: f in Y;
    then f is Permutation of Q by Th28,A1,A3;
    hence f" in Y by A2,Def37,A5;
end;

theorem Th30:
  (curry (the multF of Q)).u is Permutation of Q
proof
  set f = curry (the multF of Q).u;
  deffunc G(Element of Q) = u \ $1;
  consider g be Function of Q,Q such that
  A1: for x being Element of Q holds g.x = G(x) from FUNCT_2:sch 4;
  for x being Element of Q holds (g*f).x = (id Q).x
  proof
    let x be Element of Q;
    (g * f).x = g.(f.x) by FUNCT_2:15
    .= g.(u * x) by FUNCT_5:69
    .= G(u * x) by A1
    .= (id Q).x;
    hence thesis;
  end;
  then A2: g * f = id Q by FUNCT_2:def 8;
  A3: for x being Element of Q holds (f*g).x = (id Q).x
  proof
    let x be Element of Q;
    (f * g).x = f.(g.x) by FUNCT_2:15
    .= u * g.x by FUNCT_5:69
    .= u * G(x) by A1
    .= id Q.x;
    hence thesis;
  end;
  rng f = the carrier of Q by A3,FUNCT_2:18,def 8;
  hence thesis by FUNCT_2:57,A2, FUNCT_2:31;
end;

theorem Th31:
  (curry' (the multF of Q)).u is Permutation of the carrier of Q
proof
  set f = curry' (the multF of Q).u;
  deffunc G(Element of Q) = $1 / u;
  consider g be Function of Q,Q such that
  A1: for x being Element of Q holds g.x = G(x) from FUNCT_2:sch 4;
  for x being Element of Q holds (g*f).x = (id Q).x
  proof
    let x be Element of Q;
    (g * f).x = g.(f.x) by FUNCT_2:15
    .= g.(x * u) by FUNCT_5:70
    .= G(x * u) by A1
    .= (id Q).x;
    hence thesis;
  end;
  then A2: g * f = id Q by FUNCT_2:def 8;
  A3: for x being Element of Q holds (f*g).x = (id Q).x
  proof
    let x be Element of Q;
    (f * g).x = f.(g.x) by FUNCT_2:15
    .= g.x * u by FUNCT_5:70
    .= G(x) * u by A1
    .= (id Q).x;
    hence thesis;
  end;
  rng f = the carrier of Q by A3,FUNCT_2:18,def 8;
  hence thesis by FUNCT_2:57,A2, FUNCT_2:31;
end;

definition
  let Q;
  let H be Subset of Q;
  func Mlt H -> composition-closed inverse-closed Subset of Funcs(Q, Q) means
  :Def38:
  H left-right-mult-closed it
  &
  for X being composition-closed inverse-closed Subset of Funcs(Q,Q)
  st H left-right-mult-closed X holds it c= X;
  existence
  proof
    deffunc Phi(Subset of Funcs(Q,Q)) = MltClos1(H,$1);
    consider phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
    such that
    A1: for X being Subset of Funcs(Q,Q) holds phi.X = Phi(X)
    from FUNCT_2:sch 4;
    reconsider phi as
    c=-monotone Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
       by A1, Th26;
    set Y = lfp(Funcs(Q,Q),phi);
    A2: Y is_a_fixpoint_of phi by KNASTER:4;
    then A3: Y in dom phi & Y = phi.(Y) & phi.(Y) = Phi(Y) by ABIAN:def 3,A1;
    A4: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S by KNASTER:6;
    reconsider Y as composition-closed inverse-closed Subset of Funcs(Q,Q)
      by Th29,A1,A2,A4;
    take Y;
    thus H left-right-mult-closed Y by Th27,A1,A3;
    let S be composition-closed inverse-closed Subset of Funcs(Q,Q);
    assume A5: H left-right-mult-closed S;
    phi.S c= S
    proof
      let f be object;
      assume f in phi.S;
      then f in Phi(S) by A1;
      then PQ[Q,H,S,f] by Def37;
      hence thesis by A5,Def34,Def35;
    end;
    hence Y c= S by KNASTER:6;
  end;
  uniqueness;
end;

theorem Th32:
  for H being Subset of Q holds
  for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in Mlt H
proof
  let H be Subset of Q;
  let u be Element of Q;
  assume A1: u in H;
  H left-right-mult-closed (Mlt H) by Def38;
  hence thesis by A1;
end;

theorem Th33:
  for H being Subset of Q holds
  for u being Element of Q st u in H holds
    (curry' (the multF of Q)).u in Mlt H
proof
  let H be Subset of Q;
  let u be Element of Q;
  H left-right-mult-closed (Mlt H) by Def38;
  hence thesis;
end;

theorem Th34:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  st
  for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  Mlt H is_a_fixpoint_of phi
  &
  for S be Subset of Funcs(Q,Q) st phi.S c= S holds Mlt H c= S
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  Mlt H in bool Funcs(Q,Q)& phi is quasi_total;
  then A2: Mlt H in dom phi by FUNCT_2:def 1;
  A3: phi.(Mlt H) c= Mlt H
  proof
    let f be object;
    assume f in phi.(Mlt H);
    then f in MltClos1(H,Mlt H) by A1;
    then PQ[Q,H,Mlt H,f] by Def37;
    hence thesis by Th33,Th32,Def34,Def35;
  end;
  A4: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Mlt H c= S
  proof
    let S be Subset of Funcs(Q,Q);
    assume A5: phi.S c= S;
    set SP = {f where f is Permutation of Q : f in S};
    A6: SP c= S
    proof
      let g be object;
      assume g in SP;
      then ex f being Permutation of Q st g = f & f in S;
      hence thesis;
    end;
    S c= Funcs(the carrier of Q,the carrier of Q);
    then SP c= Funcs(the carrier of Q,the carrier of Q) by A6;
    then reconsider SP as Subset of Funcs(the carrier of Q,the carrier of Q);
    A7: for f being Element of SP st f in SP holds
      f is Permutation of the carrier of Q
    proof
      let f be Element of SP;
      assume f in SP;
      then ex g be Permutation of Q st f = g & g in S;
      hence thesis;
    end;
    for f,g being Element of SP st f in SP & g in SP holds f*g in SP
    proof
      let f,g be Element of SP;
      assume A8: f in SP & g in SP;
      reconsider f,g as Permutation of the carrier of Q by A7,A8;
      f*g in MltClos1(H,S) by Def37,A8,A6;
      then f*g in phi.S by A1;
      hence thesis by A5;
    end;
    then A9: SP is composition-closed ;
    for f being Element of SP st f in SP holds f" in SP
    proof
      let f be Element of SP;
      assume A10: f in SP;
      then f in S & f is Permutation of Q by A6,A7;
      then f" in MltClos1(H,S) by Def37;
      then A11: f" in phi.S by A1;
      reconsider f as Permutation of Q by A10,A7;
      f" is Permutation of Q;
      hence thesis by A11,A5;
    end;
    then SP is inverse-closed;
    then reconsider SP as composition-closed inverse-closed Subset of
     Funcs(Q,Q) by A9;
    for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in SP
    &
    (curry' (the multF of Q)).u in SP
    proof
      let u be Element of Q;
      assume A12: u in H;
      then (curry (the multF of Q)).u in MltClos1(H,S) by Def37;
      then A13: (curry (the multF of Q)).u in phi.(S) by A1;
      (curry (the multF of Q)).u is Permutation of Q by Th30;
      hence (curry (the multF of Q)).u in SP by A13,A5;
      (curry' (the multF of Q)).u in MltClos1(H,S) by Def37,A12;
      then A14: (curry' (the multF of Q)).u in phi.(S) by A1;
      (curry' (the multF of Q)).u is Permutation of Q by Th31;
      hence (curry' (the multF of Q)).u in SP by A14,A5;
    end;
    then H left-right-mult-closed SP;
    then Mlt H c= SP by Def38;
    hence thesis by A6;
  end;
  Mlt H c= phi.(Mlt H)
  proof
    for f,g being Element of phi.(Mlt H) st f in phi.(Mlt H) & g in phi.(Mlt H)
      holds f*g in phi.(Mlt H)
    proof
      let f,g be Element of phi.(Mlt H);
      assume A15: f in phi.(Mlt H) & g in phi.(Mlt H);
      then f is Permutation of Q & g is Permutation of Q by Th28,A1,A4,A3;
      then f * g in MltClos1(H,Mlt H) by Def37,A15,A3;
      hence thesis by A1;
    end;
    then A16: phi.(Mlt H) is composition-closed;
    for f being Element of phi.(Mlt H) st
      f in phi.(Mlt H) holds f" in phi.(Mlt H)
    proof
      let f be Element of phi.(Mlt H);
      assume A17: f in phi.(Mlt H);
      then f is Permutation of Q by A3,Th28,A1,A4;
      then f" in MltClos1(H,Mlt H) by Def37,A17,A3;
      hence thesis by A1;
    end;
    then phi.(Mlt H) is inverse-closed;
    then reconsider S = phi.(Mlt H) as composition-closed inverse-closed
    Subset of Funcs(Q,Q) by A16;
    for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in S &
    (curry' (the multF of Q)).u in S
    proof
      let u be Element of Q;
      assume u in H;
      then (curry (the multF of Q)).u in MltClos1(H,Mlt H)
      & (curry' (the multF of Q)).u in MltClos1(H,Mlt H)
      by Def37;
      hence thesis by A1;
    end;
    then H left-right-mult-closed S;
    hence thesis by Def38;
  end;
  then Mlt H = phi.(Mlt H) by A3;
  hence thesis by A4,ABIAN:def 3,A2;
end;

theorem Th35:
  for H being Subset of Q holds
  for f being Element of Funcs(Q,Q) st
  f in Mlt H holds f is Permutation of Q
proof
  let H be Subset of Q;
  deffunc Phi(Subset of Funcs(Q,Q)) = MltClos1(H,$1);
  consider phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  such that
  A1: for X being Subset of Funcs(Q,Q) holds phi.X = Phi(X)
  from FUNCT_2:sch 4;
  for S be Subset of Funcs(Q,Q) st phi.S c= S holds Mlt H c= S
  by Th34,A1;
  hence thesis by Th28,A1;
end;

definition
  let Q;
  let H be Subset of Q;
  let x be Element of Q;
  func x * H -> Subset of Q means :Def39:
  y in it iff ex h be Permutation of Q st h in Mlt H & y = h.x;
  existence
  proof
    set xH = {h.x where h is Permutation of Q : h in Mlt H};
    xH c= the carrier of Q
    proof
      let y be object;
      assume y in xH;
      then  ex h be Permutation of Q st y = h.x & h in Mlt H;
      hence thesis;
    end;
    then reconsider xH as Subset of Q;
    take xH;
    let y;
    y in xH implies ex h be Permutation of Q st h in Mlt H & y = h.x
    proof
      assume y in xH;
      then ex h being Permutation of Q st y = h.x & h in Mlt H;
      hence thesis;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let xH1,xH2 be Subset of Q;
    assume that
    A1: for y holds y in xH1 iff
    ex h be Permutation of Q st h in Mlt H & y = h.x
    and
    A2: for y holds y in xH2 iff
    ex h be Permutation of Q st h in Mlt H & y = h.x;
    for y holds y in xH1 iff y in xH2
    proof
      let y;
      y in xH1 iff
      ex h be Permutation of Q st h in Mlt H & y = h.x by A1;
      hence y in xH1 iff y in xH2 by A2;
    end;
    hence xH1 = xH2 by SUBSET_1:3;
  end;
end;

definition
  let Q;
  let H be SubLoop of Q;
  let x be Element of Q;
  func x * H -> Subset of Q equals
    x * (@ ([#] H));
  coherence;
end;

definition
  let Q;
  let N be SubLoop of Q;
  func Cosets N -> Subset-Family of Q means
  :Def41:
  for H be Subset of Q holds H in it iff ex x st H = x * N;
  existence
  proof
    set LCS = {x * N : not contradiction };
    LCS c= bool the carrier of Q
    proof
      let x be object;
      assume x in LCS;
      then ex y st x = y * N & not contradiction;
      hence thesis;
    end;
    then reconsider LCS as Subset-Family of Q;
    take LCS;
    thus thesis;
  end;
  uniqueness
  proof
    let C1,C2 be Subset-Family of Q;
    assume that
    A1: for H be Subset of Q holds H in C1 iff
    ex x st H = x * N and
    A2: for H be Subset of Q holds H in C2 iff
    ex x st H = x * N;
    thus C1 c= C2
    proof
      let H be object;
      reconsider H1=H as set by TARSKI:1;
      assume H in C1;
      then ex x st H = x * N by A1;
      hence H in C2 by A2;
    end;
    let H be object;
    reconsider H1=H as set by TARSKI:1;
    assume H in C2;
    then ex x st H = x * N by A2;
    hence H in C1 by A1;
  end;
end;

registration
  let Q;
  let N be SubLoop of Q;
  cluster Cosets N -> non empty;
  coherence
  proof
    1.Q * N in Cosets N by Def41;
    hence thesis;
  end;
end;

begin :: Normal Subloop

::We define the notion of a normal subloop
::and construct quotients by normal subloops.

definition
  let Q be multLoopStr;
  let H1,H2 be Subset of Q;
  func H1 * H2 -> Subset of Q means
  :Def42:
  for x being Element of Q holds
  x in it iff
  ex y,z be Element of Q st y in H1 & z in H2 & x = y * z;
  existence
  proof
    set H3 = {x where x is Element of Q : ex y,z be Element of Q st y in H1 &
      z in H2 & x = y * z};
    H3 c= the carrier of Q
    proof
      let x be object;
      assume x in H3;
      then ex x1 be Element of Q st x = x1 & ex y,z be Element of Q st
        y in H1 & z in H2 & x1 = y * z;
      hence thesis;
    end;
    then reconsider H3 as Subset of Q;
    take H3;
    let x be Element of Q;
    x in H3 implies ex y,z be Element of Q st y in H1 & z in H2 & x = y * z
    proof
      assume x in H3;
      then consider x1 be Element of Q such that
      A1: x = x1 & ex y,z be Element of Q st y in H1 & z in H2 & x1 = y * z;
      thus thesis by A1;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let H31,H32 be Subset of Q;
    assume that
    A2: for x being Element of Q holds
    x in H31 iff
    ex y,z be Element of Q st y in H1 & z in H2 & x = y * z
    and
    A3: for x being Element of Q holds
    x in H32 iff
    ex y,z be Element of Q st y in H1 & z in H2 & x = y * z;
    now
      let x be Element of Q;
      x in H31 iff
      ex y,z be Element of Q st y in H1 & z in H2 & x = y * z by A2;
      hence x in H31 iff x in H32 by A3;
    end;
    hence thesis by SUBSET_1:3;
  end;
  func H1 \ H2 -> Subset of Q means
  for x being Element of Q holds
  x in it iff
  ex y,z be Element of Q st y in H1 & z in H2 & x = y \ z;
  existence
  proof
    set H3 = {x where x is Element of Q : ex y,z be Element of Q st y in H1 &
      z in H2 & x = y \ z};
    H3 c= the carrier of Q
    proof
      let x be object;
      assume x in H3;
      then ex x1 be Element of Q st x = x1 & ex y,z be Element of Q st
        y in H1 & z in H2 & x1 = y \ z;
      hence thesis;
    end;
    then reconsider H3 as Subset of Q;
    take H3;
    let x be Element of Q;
    x in H3 implies ex y,z be Element of Q st y in H1 & z in H2 & x = y \ z
    proof
      assume x in H3;
      then consider x1 be Element of Q such that
      A4: x = x1 & ex y,z be Element of Q st y in H1 & z in H2 & x1 = y \ z;
      thus thesis by A4;
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let H31,H32 be Subset of Q;
    assume that
    A5: for x being Element of Q holds
    x in H31 iff
    ex y,z be Element of Q st y in H1 & z in H2 & x = y \ z
    and
    A6: for x being Element of Q holds
    x in H32 iff
    ex y,z be Element of Q st y in H1 & z in H2 & x = y \ z;
    now
      let x be Element of Q;
      x in H31 iff
      ex y,z be Element of Q st y in H1 & z in H2 & x = y \ z by A5;
      hence x in H31 iff x in H32 by A6;
    end;
    hence thesis by SUBSET_1:3;
  end;
end;

definition
  let Q be multLoop;
  let H be SubLoop of Q;
  attr H is normal means
  :Def44:
  for x,y being Element of Q holds
  (x * H) * (y * H) = (x * y) * H
  &
  for z being Element of Q holds
  ((x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H))
  &
  ((y * H) * (x * H) = (z * H) * (x * H) implies (y * H) = (z * H));
end;

registration
  let Q;
  cluster normal for SubLoop of Q;
  existence
  proof
    reconsider Q1=Q as non empty multLoopStr;
    A1: the multF of Q1 = (the multF of Q)||the carrier of Q1;
    the OneF of Q1 = the OneF of Q;
    then reconsider Q1 as SubLoop of Q by A1,Def30;
    take Q1;
    A2: for x,y being Element of Q holds y in x * Q1
    proof
      let x,y be Element of Q;
      ex g being Permutation of Q st g in Mlt (@ ([#] Q1)) & y = g.x
      proof
        reconsider g = (curry (the multF of Q)).(y / x)
        as Permutation of Q by Th30;
        A3: (@ ([#] Q1)) left-right-mult-closed Mlt (@ ([#] Q1)) by Def38;
        g.x  = (y / x) * x by FUNCT_5:69 .= y;
        hence thesis by A3;
      end;
      hence y in x * Q1 by Def39;
    end;
    A5: for x,y being Element of Q holds x * Q1 = y * Q1
    proof
      let x,y be Element of Q;
      for v being Element of Q holds v in x * Q1 iff v in y * Q1 by A2;
      hence thesis by SUBSET_1:3;
    end;
    now
      let x,y be Element of Q;
      for v being Element of Q holds v in (x*Q1)*(y*Q1) iff v in (x*y)*Q1
      proof
        let v be Element of Q;
        thus v in (x*Q1)*(y*Q1) implies v in (x*y)*Q1 by A2;
        assume v in (x*y)*Q1;
        ex u,w st u in x * Q1 & w in y * Q1 & v = u * w
        proof
          take v,1.Q;
          thus thesis by A2;
        end;
        hence v in (x*Q1)*(y*Q1) by Def42;
      end;
      hence (x*Q1)*(y*Q1) = (x*y)*Q1 by SUBSET_1:3;
      let z;
      thus (x*Q1)*(z*Q1) = (y*Q1)*(z*Q1) implies (x*Q1) = (y*Q1) by A5;
      thus (z*Q1)*(x*Q1) = (z*Q1)*(y*Q1) implies (x*Q1) = (y*Q1) by A5;
    end;
    hence thesis;
  end;
end;

definition
  let Q;
  let N be normal SubLoop of Q;
  func SubLoop_As_Coset N -> Element of Cosets N equals
  1.Q * N;
  coherence by Def41;
end;

definition
  let Q;
  let N be normal SubLoop of Q;
  func Coset_Loop_Op N -> BinOp of Cosets N
  means
  :Def46:
  for H1,H2 be Element of Cosets N holds
  it.(H1,H2) = H1 * H2;
  existence
  proof
    deffunc G(Element of Cosets N,Element of Cosets N) = $1 * $2;
    A1: for H1,H2 being Element of Cosets N holds G(H1,H2) in Cosets N
    proof
      let H1,H2 be Element of Cosets N;
       consider x being Element of Q such that
      A2: H1 = x * N
      by Def41;
      consider y being Element of Q such that
      A3: H2 = y * N
      by Def41;
      G(H1,H2) = (x * y) * N by Def44,A2,A3;
      hence G(H1,H2) in Cosets N by Def41;
    end;
    consider g being Function of [: Cosets N,Cosets N :],Cosets N such that
    A4: for H1,H2 being Element of Cosets N holds g.(H1,H2) = G(H1,H2)
    from FUNCT_7:sch 1(A1);
    take g;
    thus thesis by A4;
  end;
  uniqueness
  proof
    let LCL1,LCL2 be BinOp of Cosets N such that
    A5:  for H1,H2 being Element of Cosets N holds
    LCL1.(H1,H2) = H1 * H2 and
    A6:  for H1,H2 being Element of Cosets N holds
    LCL2.(H1,H2) = H1 * H2;
    for H1,H2 being Element of Cosets N holds LCL1.(H1,H2) = LCL2.(H1,H2)
    proof
      let H1,H2 be Element of Cosets N;
      LCL1.(H1,H2) = H1 * H2 by A5
      .= LCL2.(H1,H2) by A6;
      hence thesis;
    end;
    hence thesis;
  end;
end;

definition
  let Q;
  let N be normal SubLoop of Q;
  func Q _/_ N -> strict multLoopStr equals
    multLoopStr(#Cosets N,Coset_Loop_Op N,SubLoop_As_Coset N#);
  coherence;
end;

registration
  let Q;
  let N be normal SubLoop of Q;
  cluster Q _/_ N -> non empty;
  coherence;
end;

registration
  let Q;
  let N be normal SubLoop of Q;
  cluster Q _/_ N -> well-unital invertible cancelable;
  coherence
proof
  set QN = Q _/_ N;
A1: for H being Element of QN holds H * 1.QN = H & 1.QN * H = H
  proof
    let H be Element of QN;
    H in Cosets N;
    then consider x being Element of Q such that
    A2: H = x * N by Def41;
    A3: H * 1.QN = H
    proof
      reconsider H as Element of Cosets N;
      (the multF of QN).(H,1.QN) = H * (1.Q * N) by Def46
      .= (x * 1.Q) * N by A2,Def44
      .= H by A2;
      hence thesis;
    end;
    1.QN * H = H
    proof
      reconsider H as Element of Cosets N;
      (the multF of QN).(1.QN,H) = (1.Q * N) * H by Def46
      .= (1.Q * x) * N by A2,Def44
      .= H by A2;
      hence thesis;
    end;
    hence thesis by A3;
  end;
  A4: for H1,H2 being Element of QN holds
  ex H3 being Element of QN st H1 * H3 = H2
  proof
    let H1,H2 be Element of QN;
    H1 in Cosets N;
    then consider x being Element of Q such that
    A5: H1 = x * N by Def41;
    H2 in Cosets N;
    then consider y being Element of Q such that
    A6: H2 = y * N by Def41;
    reconsider H3 = (x \ y) * N as Element of QN by Def41;
    take H3;
    (the multF of QN).(H1,H3) = (x * N) * ((x \ y) * N) by A5,Def46
    .= (x * (x \ y)) * N by Def44
    .= H2 by A6;
    hence thesis;
  end;
  A7: for H1,H2 being Element of QN holds
  ex H3 being Element of QN st H3 * H1 = H2
  proof
    let H1,H2 be Element of QN;
    H1 in Cosets N;
    then consider x being Element of Q such that
    A8: H1 = x * N by Def41;
    H2 in Cosets N;
    then consider y being Element of Q such that
    A9: H2 = y * N by Def41;
    reconsider H3 = (y / x) * N as Element of QN by Def41;
    take H3;
    (the multF of QN).(H3,H1) = ((y / x) * N) * (x * N) by A8,Def46
    .= ((y / x) * x) * N by Def44
    .= H2 by A9;
    hence thesis;
  end;
  for H1 being Element of QN holds H1 is left_mult-cancelable
  proof
    let H1 be Element of QN;
    for H2,H3 being Element of QN st H1*H2 = H1*H3 holds H2 = H3
    proof
      let H2,H3 be Element of QN;
      H1 in Cosets N;
      then consider x being Element of Q such that
      A10: H1 = x * N by Def41;
      H2 in Cosets N;
      then consider y being Element of Q such that
      A11: H2 = y * N by Def41;
      H3 in Cosets N;
      then consider z being Element of Q such that
      A12: H3 = z * N by Def41;
      assume A13: H1*H2 = H1*H3;
      (x*N)*(y*N) = H1 * H2 by A10,A11,Def46
      .= (x*N)*(z*N) by A10,A12,A13,Def46;
      hence thesis by A11,A12,Def44;
    end;
    hence thesis by ALGSTR_0:def 20;
  end;
  then A14: QN is left_mult-cancelable by ALGSTR_0:def 23;
  for H1 being Element of QN holds H1 is right_mult-cancelable
  proof
    let H1 be Element of QN;
    let H2,H3 be Element of QN;
    H1 in Cosets N;
    then consider x being Element of Q such that
    A15: H1 = x * N by Def41;
    H2 in Cosets N;
    then consider y being Element of Q such that
    A16: H2 = y * N by Def41;
    H3 in Cosets N;
    then consider z being Element of Q such that
    A17: H3 = z * N by Def41;
    assume A18: H2*H1 = H3*H1;
    (y*N)*(x*N) = H2 * H1 by A15,A16,Def46
    .= (z*N)*(x*N) by A18,A15,A17,Def46;
    hence thesis by A16,A17,Def44;
  end;
  then QN is right_mult-cancelable by ALGSTR_0:def 24;
  hence thesis by A1,A7,ALGSTR_1:def 6,A4,A14;
end;
end;

definition
  let Q;
  let N be normal SubLoop of Q;
  func QuotientHom(Q,N) -> Function of Q,Q _/_ N means
  :Def48:
  for x holds it.x = x * N;
  existence
  proof
    deffunc F(Element of Q) = $1 * N;
    consider f be Function of Q,bool the carrier of Q
    such that
    A1: for x being Element of Q holds f.x = F(x)
    from FUNCT_2:sch 4;
    A2: dom f = the carrier of Q by FUNCT_2:def 1;
    A3: rng f c= the carrier of Q _/_ N
    proof
      let H be object;
      assume H in rng f;
      then consider x being object such that
      A4: x in dom f & H = f.x by FUNCT_1:def 3;
      reconsider x as Element of Q by A4,FUNCT_2:def 1;
      H = x * N by A1,A4;
      hence thesis  by Def41;
    end;
    reconsider f as Function of Q,Q _/_ N by FUNCT_2:2,A2,A3;
    take f;
    thus thesis by A1;
  end;
  uniqueness
  proof
    let f,g be Function of Q,Q _/_ N such that
    A5: for x holds f.x = x * N
    and
    A6: for x holds g.x = x * N;
    let x;
    thus f.x = x * N by A5
    .= g.x by A6;
  end;
end;

registration
  let Q;
  let N be normal SubLoop of Q;
  cluster QuotientHom(Q,N) -> homomorphic;
  coherence
  proof
    set f = QuotientHom(Q,N);
    thus f.(1.Q) = 1.(Q _/_ N) by Def48;
    let x,y be Element of Q;
    reconsider xN = x * N as Element of Cosets N by Def41;
    reconsider yN = y * N as Element of Cosets N by Def41;
    f.(x * y) = (x * y) * N by Def48
    .= (x * N) * (y * N) by Def44
    .= (Coset_Loop_Op N).(xN,yN) by Def46
    .= (the multF of Q _/_ N).(f.x,yN) by Def48
    .= (the multF of Q _/_ N).(f.x,f.y) by Def48;
    hence thesis;
  end;
end;

theorem Th36:
  for H being SubLoop of Q holds
  for x,y holds
  for x1,y1 being Element of H st
  x = x1 & y = y1
  holds
  x * y = x1 * y1
proof
  let H be SubLoop of Q;
  let x,y;
  let x1,y1 be Element of H;
  assume A1: x = x1 & y = y1;
  x1 * y1 = ((the multF of Q)||the carrier of H).(x1,y1) by Def30
  .= x * y by A1,RING_3:1;
  hence thesis;
end;

theorem Th37:
  for H being SubLoop of Q holds
  for x,y st x in the carrier of H & y in the carrier of H holds
    x * y in the carrier of H
proof
  let H be SubLoop of Q;
  let x,y;
  assume
  x in the carrier of H &
   y in the carrier of H;
  then reconsider x1 = x,y1=y  as Element of H;
  x * y = x1 * y1 by Th36;
  hence thesis;
end;

theorem Th38:
  for H being SubLoop of Q holds
  for x,y holds
  for x1,y1 being Element of H st
  x = x1 & y = y1
  holds
  x \ y = x1 \ y1
proof
  let H be SubLoop of Q;
  let x,y;
  let x1,y1 be Element of H;
  assume A1: x = x1 & y = y1;
   the carrier of H c= the carrier of Q by Def30;
  then reconsider x1y1 = x1 \ y1 as Element of Q;
  x * x1y1 = x1 * (x1 \ y1) by Th36,A1
  .= y by A1;
  hence thesis;
end;

theorem Th39:
  for H being SubLoop of Q holds
  for x,y st x in the carrier of H & y in the carrier of H holds
    x \ y in the carrier of H
proof
  let H be SubLoop of Q, x,y such that
  A1: x in the carrier of H & y in the carrier of H;
  reconsider x1 = x,y1=y  as Element of H by A1;
  x \ y = x1 \ y1 by Th38;
  hence thesis;
end;

theorem Th40:
  for H being SubLoop of Q holds
  for x,y holds
  for x1,y1 being Element of H st
  x = x1 & y = y1
  holds
  x / y = x1 / y1
proof
  let H be SubLoop of Q,x,y;
  let x1,y1 be Element of H;
  the carrier of H c= the carrier of Q by Def30;
  then reconsider x1y1 = x1 / y1 as Element of Q;
  assume A1: x = x1 & y = y1;
  then x1y1 * y = (x1 / y1) * y1 by Th36
  .= x by A1;
  hence thesis;
end;

theorem Th41:
  for H being SubLoop of Q holds
  for x,y st x in the carrier of H & y in the carrier of H holds
    x / y in the carrier of H
proof
  let H be SubLoop of Q,x,y;
  assume x in the carrier of H & y in the carrier of H;
  then reconsider x1 = x,y1=y as Element of H;
  x / y = x1 / y1 by Th40;
  hence thesis;
end;

scheme MltInd {Q() -> multLoop, H() -> Subset of Q(), P[Function of Q(),Q()]}:
  for f being Function of Q(),Q() st f in Mlt H() holds P[f]
  provided
  A1: for u being Element of Q() st u in H() holds
  for f being Function of Q(),Q() st for x being Element of Q() holds
    f.x = x * u holds P[f]
  and
  A2: for u being Element of Q() st u in H() holds
  for f being Function of Q(),Q() st for x being Element of Q() holds
    f.x = u * x holds P[f]
  and
  A3: for g,h being Permutation of Q() st P[g] & P[h] holds P[g*h]
  and
  A4: for g being Permutation of Q() st P[g] holds P[g"]
proof
  deffunc Phi(Subset of Funcs(Q(),Q())) = MltClos1(H(),$1);
  consider phi be Function of bool Funcs(Q(),Q()),bool Funcs(Q(),Q())
  such that
  A5: for X being Subset of Funcs(Q(),Q()) holds phi.X = Phi(X)
  from FUNCT_2:sch 4;
  set SP = {f where f is Function of Q(),Q() : P[f]};
  SP c= Funcs(Q(),Q())
  proof
    let f be object;
    assume f in SP;
    then ex g being Function of Q(),Q() st f = g & P[g];
    hence thesis by FUNCT_2:9;
  end;
  then reconsider SP as Subset of Funcs(Q(),Q());
  phi.(SP) c= SP
  proof
    let f be object;
    assume f in phi.(SP);
    then f in MltClos1(H(),SP) by A5;
    then per cases by Def37;
    suppose ex u be Element of Q() st u in H() &
      f = (curry' (the multF of Q())).u;
      then consider u being Element of Q() such that
      A6: u in H() & f = (curry' (the multF of Q())).u;
      reconsider f as Function of Q(),Q() by A6;
      for x being Element of Q() holds f.x = x * u by A6,FUNCT_5:70;
      then P[f] by A1,A6;
      hence thesis;
    end;
    suppose ex u be Element of Q() st u in H() &
      f = (curry (the multF of Q())).u;
      then consider u being Element of Q() such that
      A7: u in H() & f = (curry (the multF of Q())).u;
     reconsider f as Function of Q(),Q() by A7;
      for x being Element of Q() holds f.x = u * x by A7,FUNCT_5:69;
      then P[f] by A2,A7;
      hence thesis;
    end;
    suppose ex g,h be Permutation of Q() st g in SP & h in SP & f = g*h;
      then consider g,h being Permutation of Q() such that
      A8: g in SP & h in SP & f = g*h;
        consider g2 being Function of Q(),Q() such that
        A9: g = g2 & P[g2] by A8;
     A10: ex h2 be Function of Q(),Q() st
        h = h2 & P[h2] by A8;
      P[g*h] by A10,A3,A9;
      hence thesis by A8;
    end;
    suppose ex g be Permutation of Q() st g in SP & f = g";
      then consider g being Permutation of Q() such that
      A11: g in SP & f = g";
      ex g2 being Function of Q(),Q() st g = g2 & P[g2] by A11;
      then P[g"] by A4;
      hence thesis by A11;
    end;
  end;
  then A12: Mlt H() c= SP by Th34,A5;
  let f be Function of Q(),Q();
  assume f in Mlt H();
  then f in SP by A12;
  then ex g be Function of Q(),Q() st f = g & P[g];
  hence thesis;
end;

theorem Th42:
  for N being SubLoop of Q holds
  for f being Function of Q,Q st f in Mlt (@ ([#] N)) holds
  for x holds x in (@ ([#] N)) iff f.x in (@ ([#] N))
proof
  let N be SubLoop of Q;
  reconsider H = @ ([#] N) as Subset of Q;
  defpred P[Function of Q,Q] means for x holds x in H iff $1.x in H;
  A1: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = x * u holds P[f]
  proof
    let u;
    assume A2:u in H;
    let f be Function of Q,Q;
    assume A3: for x holds f.x = x * u;
    P[f]
    proof
      let x;
      thus x in H implies f.x in H
      proof
        assume x in H;
        then x * u in the carrier of N by Th37,A2;
        hence thesis by A3;
      end;
      assume f.x in H;
      then reconsider xu1 = x * u as Element of N by A3;
      reconsider u1 = u as Element of N by A2;
      the carrier of N c= the carrier of Q by Def30;
      then reconsider xu1u1 = xu1 / u1 as Element of Q;
      A4: x * u = (xu1 / u1) * u1
      .= xu1u1 * u by Th36;
      x = (xu1u1 * u) / u by A4
      .= (xu1 / u1);
      hence thesis;
    end;
    hence thesis;
  end;
  A5: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = u * x holds P[f]
  proof
    let u;
    assume A6: u in H;
    let f be Function of Q,Q;
    assume A7: for x holds f.x = u * x;
    P[f]
    proof
      let x;
      thus x in H implies f.x in H
      proof
        assume x in H;
        then u * x in the carrier of N by Th37,A6;
        hence thesis by A7;
      end;
      assume f.x in H;
      then reconsider ux1 = u * x,u1=u as Element of N by A7,A6;
      the carrier of N c= the carrier of Q by Def30;
      then reconsider u1ux1 = u1 \ ux1 as Element of Q;
      u * x = u1 * (u1 \ ux1)
      .= u * u1ux1 by Th36;
      then x  = u \ (u * u1ux1)
      .= (u1 \ ux1);
      hence thesis;
    end;
    hence thesis;
  end;
  A8: for g,h being Permutation of Q st P[g] & P[h] holds P[g*h]
  proof
    let g,h be Permutation of Q such that A9: P[g] & P[h];
    let x;
    x in H iff h.x in H by A9;
    then x in H iff g.(h.x) in H by A9;
    hence thesis by FUNCT_2:15;
  end;
  A10: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of Q such that A11: P[g];
    let x;
    x = (id the carrier of Q).x
    .= (g*(g")).x by FUNCT_2:61
    .= g.((g").x) by FUNCT_2:15;
    hence x in H iff (g").x in H by A11;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A5,A8,A10);
  hence thesis;
end;

theorem Th43:
  for N being normal SubLoop of Q holds
  the carrier of N = 1.Q * N
proof
  let N be normal SubLoop of Q;
  A1: the carrier of N c= the carrier of Q by Def30;
  thus the carrier of N c= 1.Q * N
  proof
    let x be object;
    assume A2: x in the carrier of N;
    then reconsider x as Element of Q by A1;
    A3: (curry (the multF of Q)).x in Mlt (@ ([#] N)) by Th32,A2;
    reconsider h = (curry (the multF of Q)).x as Permutation of Q
      by Th30;
    h.(1.Q) = x * 1.Q by FUNCT_5:69;
    hence thesis by Def39,A3;
  end;
  let x be object;
  assume x in 1.Q * N;
  then A4:ex h be Permutation of Q st
  h in Mlt (@ [#] N) & x = h.(1.Q) by Def39;
  1.N = 1.Q by Def30;
  hence thesis by Th42,A4;
end;


theorem Th44:
  for N being normal SubLoop of Q holds
  Ker (QuotientHom(Q,N)) = @ ([#] N)
proof
  let N be normal SubLoop of Q;
  A1: the carrier of N c= the carrier of Q by Def30;
  set f = QuotientHom(Q,N);
  for x holds x in Ker f iff x in @ ([#] N)
  proof
    let x;
    thus x in Ker f implies x in @ ([#] N)
    proof
      assume A2: x in Ker f;
      A3:x * N = f.x by Def48
      .= 1.(Q _/_ N) by Def29,A2
      .= 1.Q * N;
      A4: 1.N = 1.Q by Def30;
      reconsider h = (curry (the multF of Q)).(1.Q) as Permutation of Q
        by Th30;
      A5: h in Mlt (@ [#] N) by A4,Th32;
      A6: h.x in x * (@ ([#] N)) by Def39,A5;
      h.x = 1.Q * x by FUNCT_5:69;
      hence thesis by A6,A3,Th43;
    end;
    assume A7: x in @ ([#] N);
    A8: for y holds y in x * N iff y in 1.Q * N
    proof
      let y;
      thus y in x * N implies y in 1.Q * N
      proof
        assume y in x * N;
        then consider h being Permutation of Q such that
        A9: h in Mlt (@ ([#] N)) & h.x = y by Def39;
        h.x in @ ([#] N) by Th42,A9,A7;
        hence thesis by A9,Th43;
      end;
      assume y in 1.Q * N;
      then reconsider y1 = y as Element of N by Th43;
      reconsider x1 = x as Element of N by A7;
      ex h being Permutation of Q st h in Mlt (@ ([#] N)) & y = h.x
      proof
        reconsider y1x1 = y1 / x1 as Element of Q by A1;
        reconsider h = (curry (the multF of Q)).(y1x1) as Permutation of Q
        by Th30;
        take h;
        thus h in Mlt (@ [#] N) by Th32;
        h.x = y1x1 * x by FUNCT_5:69
        .= (y / x) * x by Th40
        .= y;
        hence h.x = y;
      end;
      hence thesis by Def39;
    end;
    f.x = x * N by Def48
    .= 1.(Q _/_ N) by A8,SUBSET_1:3;
    hence thesis by Def29;
  end;
  hence thesis by SUBSET_1:3;
end;

theorem Th45:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for h being Function of Q,Q st h in Mlt (Ker f) holds
  f*h = f
proof
  let Q2 be multLoop;
  let f be homomorphic Function of Q,Q2;
  set H = Ker f;
  defpred P[Function of Q,Q] means f * $1 = f;
  A1: for u being Element of Q st u in H holds
  for h being Function of Q,Q st for x being Element of Q holds
    h.x = x * u holds P[h]
  proof
    let u;
    assume A2: u in H;
    let h be Function of Q,Q;
    assume A3: for x holds h.x = x * u;
    P[h]
    proof
      for x holds (f*h).x = f.x
      proof
        let x;
        thus (f*h).x = f.(h.x) by FUNCT_2:15
        .= f.(x * u) by A3
        .= f.x * f.u by Def28b
        .= f.x * 1.Q2 by A2,Def29
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  A4: for u being Element of Q st u in H holds
  for h being Function of Q,Q st for x being Element of Q holds
    h.x = u * x holds P[h]
  proof
    let u;
    assume A5: u in H;
    let h be Function of Q,Q;
    assume A6: for x holds h.x = u * x;
    P[h]
    proof
      for x holds (f*h).x = f.x
      proof
        let x;
        thus (f*h).x = f.(h.x) by FUNCT_2:15
        .= f.(u * x) by A6
        .= f.u * f.x by Def28b
        .= 1.Q2 * f.x by A5,Def29
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  A7: for g,h being Permutation of Q st P[g] & P[h] holds P[g*h] by RELAT_1:36;
  A8: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of Q such that A9: P[g];
    P[g"]
    proof
      for x holds (f*(g")).x = f.x
      proof
        let x;
        thus (f*(g")).x = f.((g").x) by FUNCT_2:15
        .= f.(g.((g").x)) by FUNCT_2:15,A9
        .= f.((g*g").x) by FUNCT_2:15
        .= f.((id the carrier of Q).x) by FUNCT_2:61
        .= f.x;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    hence thesis;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A4,A7,A8);
  hence thesis;
end;

theorem Th46:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for x,y holds
  y in x * Ker f iff f.x = f.y
proof
  let Q2 be multLoop,f be homomorphic Function of Q,Q2,x,y;
  thus y in x * Ker f implies f.x = f.y
  proof
    assume y in x * Ker f;
    then consider h being Permutation of Q such that
    A1: h in Mlt (Ker f) & y = h.x by Def39;
    f.x = (f*h).x by Th45,A1
    .= f.y by A1,FUNCT_2:15;
    hence thesis;
  end;
  assume A2: f.x = f.y;
  ex h being Permutation of Q st h in Mlt (Ker f) & y = h.x
  proof
    reconsider h = (curry (the multF of Q)).(y / x)
    as Permutation of Q by Th30;
    take h;
    f.(y / x) = f.y / f.x by Th14
    .= 1.Q2 by Th6,A2;
    then A3: y / x in Ker f by Def29;
    h.x = (y / x) * x by FUNCT_5:69
    .= y;
    hence thesis by A3,Th32;
  end;
  hence thesis by Def39;
end;

theorem Th47:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for x,y holds
  y in x * lp (Ker f) iff f.x = f.y
proof
  let Q2 be multLoop,f be homomorphic Function of Q,Q2,x,y;
  y in x * lp (Ker f) iff y in x * Ker f by Th19;
  hence y in x * lp (Ker f) iff f.x = f.y by Th46;
end;

theorem Th48:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for x,y holds
  x * lp (Ker f) = y * lp (Ker f) iff f.x = f.y
proof
  let Q2 be multLoop,f be homomorphic Function of Q,Q2;
  A1: for x,y holds f.x = f.y implies x * lp (Ker f) c= y * lp (Ker f)
  proof
    let x,y such that A2: f.x = f.y;
    let z be object;
    assume A3: z in x * lp (Ker f);
    then f.x = f.z by Th47;
    hence z in y * lp (Ker f) by A3,A2,Th47;
  end;
  let x,y;
  x * lp (Ker f) = y * lp (Ker f) implies f.x = f.y
  proof
    assume A4: x * lp (Ker f) = y * lp (Ker f);
    f.y = f.y;
    then y in y * lp (Ker f) by Th47;
    hence thesis by A4,Th47;
  end;
  hence thesis by A1;
end;

theorem
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  lp (Ker f) is normal
proof
  let Q2 be multLoop;
  let f be homomorphic Function of Q,Q2;
  set H = lp (Ker f);
  A1: for x,y holds (x * H) * (y * H) = (x * y) * H
  proof
    let x,y;
    for z holds z in (x * H) * (y * H) iff z in (x * y) * H
    proof
      let z;
      thus z in (x * H) * (y * H) implies z in (x * y) * H
      proof
        assume z in (x * H) * (y * H);
        then consider v,w such that
        A2: v in x * H & w in y * H & z = v * w by Def42;
        A3: f.y = f.w by Th47,A2;
        f.z = f.v * f.w by Def28b,A2
        .= f.x * f.y by Th47,A2,A3
        .= f.(x * y) by Def28b;
        hence z in (x * y) * H by Th47;
      end;
      assume z in (x * y) * H;
      then A4: f.z = f.(x * y) by Th47;
      ex v,w st v in x * H & w in y * H & z = v * w
      proof
        take z / y,y;
        A5: f.(z / y) = f.z / f.y by Th14
        .= (f.x * f.y) / f.y by A4,Def28b
        .= f.x;
        f.y = f.y;
        hence thesis by A5,Th47;
      end;
      hence z in (x * H) * (y * H) by Def42;
    end;
    hence thesis by SUBSET_1:3;
  end;
  for x,y holds
  (x * H) * (y * H) = (x * y) * H
  &
  for z holds
  ((x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H))
  &
  ((y * H) * (x * H) = (z * H) * (x * H) implies (y * H) = (z * H))
  proof
    let x,y;
    thus (x * H) * (y * H) = (x * y) * H by A1;
    let z;
    thus (x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H)
    proof
      assume (x * H) * (y * H) = (x * H) * (z * H);
      then (x * y) * H = (x * H) * (z * H) by A1;
      then A6: (x * y) * H = (x * z) * H by A1;
      f.y = f.x \ (f.x * f.y)
      .= f.x \ f.(x * y) by Def28b
      .= f.x \ f.(x * z) by A6,Th48
      .= f.x \ (f.x * f.z) by Def28b
      .= f.z;
      hence (y * H) = (z * H) by Th48;
    end;
    assume (y * H) * (x * H) = (z * H) * (x * H);
    then (y * x) * H = (z * H) * (x * H) by A1;
    then A7: (y * x) * H = (z * x) * H by A1;
    f.y = (f.y * f.x) / f.x
    .= f.(y * x) / f.x by Def28b
    .= f.(z * x) / f.x by A7,Th48
    .= (f.z * f.x) / f.x by Def28b
    .= f.z;
    hence (y * H) = (z * H) by Th48;
  end;
  hence thesis;
end;

theorem Th50:
  1.Q in [#] (lp (Cent Q)) & 1.Q in Cent Q
proof
  the OneF of (lp (Cent Q)) = 1.Q by Def30;
  then 1.Q in [#] (lp (Cent Q));
  hence thesis by Th25;
end;

theorem Th51:
  for f being Function of Q,Q st f in Mlt (Cent Q) holds
  ex z st z in Cent Q & for x holds f.x = x * z
proof
  set H = Cent Q;
  defpred P[Function of Q,Q] means ex z st z in H & for x holds $1.x = x * z;
  A1: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = x * u holds P[f];
  A2: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = u * x holds P[f]
  proof
    let u;
    assume A3: u in H;
    then A4: u in Comm Q by XBOOLE_0:def 4;
    let f be Function of Q,Q;
    assume A5: for x holds f.x = u * x;
    P[f]
    proof
      take u;
      thus u in Cent Q by A3;
      let x;
      f.x = u * x by A5
      .= x * u by Def25,A4;
      hence thesis;
    end;
    hence thesis;
  end;
  A6: for g,h being Permutation of Q st P[g] & P[h] holds P[g*h]
  proof
    let g,h be Permutation of Q;
    assume A7: P[g] & P[h];
    consider u such that
    A8: u in H & for x holds g.x = x * u by A7;
    consider v such that
    A9: v in H & for x holds h.x = x * v by A7;
    take (v * u);
    u in [#] (lp (Cent Q)) & v in [#] (lp (Cent Q)) by Th25,A8,A9;
    then v * u in [#] (lp (Cent Q)) by Th37;
    hence v * u in H by Th25;
    u in Nucl Q by A8,XBOOLE_0:def 4;
    then A10: u in Nucl_r Q by Th12;
    let x;
    (g*h).x = g.(h.x) by FUNCT_2:15
    .= g.(x * v) by A9
    .= (x * v) * u by A8
    .= x * (v * u) by A10,Def24;
    hence thesis;
  end;
  A11: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of  Q;
    assume P[g];
    then consider v such that
    A12: v in H & for x holds g.x = x * v;
    v in Nucl Q by A12,XBOOLE_0:def 4;
    then A13: v in Nucl_m Q by Th12;
    P[g"]
    proof
      take (v \ 1.Q);
      A14: 1.Q in [#] (lp (Cent Q)) by Th50;
      v in [#] (lp (Cent Q)) by Th25,A12;
      then v \ 1.Q in [#] (lp (Cent Q)) by Th39,A14;
      hence v \ 1.Q in Cent Q by Th25;
      let x;
      reconsider h = (curry' (the multF of Q)).(v \ 1.Q)
      as Permutation of Q by Th31;
      for y holds (h*g).y = (id Q).y
      proof
        let y;
        (h*g).y = h.(g.y) by FUNCT_2:15
        .= h.(y * v) by A12
        .= (y * v) * (v \ 1.Q) by FUNCT_5:70
        .= y * (v * (v \ 1.Q)) by Def23,A13
        .= y;
        hence thesis;
      end;
      then (g").x = h.x by FUNCT_2:60,def 8
      .= x * (v \ 1.Q) by FUNCT_5:70;
      hence thesis;
    end;
    hence thesis;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A2,A6,A11);
  hence thesis;
end;

theorem Th52:
  y in x * lp (Cent Q) iff ex z st z in Cent Q & y = x * z
proof
  thus y in x * lp (Cent Q) implies ex z st z in Cent Q & y = x * z
  proof
    assume y in x * lp (Cent Q);
    then y in x * Cent Q by Th25;
    then consider h being Permutation of Q such that
    A1: h in Mlt (Cent Q) & h.x = y by Def39;
    consider z such that
    A2: z in Cent Q & for v holds h.v = v * z by Th51,A1;
    take z;
    thus thesis by A2,A1;
  end;
  given z such that
  A3: z in Cent Q & y = x * z;
  reconsider h = (curry' (the multF of Q)).(z) as Permutation of Q
  by Th31;
  ex h being Permutation of Q st h in Mlt (Cent Q) & h.x = y
  proof
    reconsider h = (curry' (the multF of Q)).(z) as Permutation of Q
    by Th31;
    take h;
    thus thesis by FUNCT_5:70,Th33,A3;
  end;
  then y in x * Cent Q by Def39;
  hence thesis by Th25;
end;

theorem Th53:
  x * lp (Cent Q) = y * lp (Cent Q) iff ex z st z in Cent Q & y = x * z
proof
  thus x * lp (Cent Q) = y * lp (Cent Q) implies
    ex z st z in Cent Q & y = x * z
  proof
    assume A1: x * lp (Cent Q) = y * lp (Cent Q);
    1.Q in Cent Q & y = y * 1.Q by Th50;
    hence ex z st z in Cent Q & y = x * z by A1,Th52;
  end;
  thus (ex z st z in Cent Q & y = x * z) implies
    x * lp (Cent Q) = y * lp (Cent Q)
  proof
    assume ex z st z in Cent Q & y = x * z;
    then consider z such that
    A2: z in Cent Q & y = x * z;
    z in Nucl Q by A2,XBOOLE_0:def 4;
    then A3: z in Nucl_m Q by Th12;
    for w holds w in x * lp (Cent Q) iff w in y * lp (Cent Q)
    proof
      let w;
      thus w in x * lp (Cent Q) implies w in y * lp (Cent Q)
      proof
        assume w in x * lp (Cent Q);
        then consider v such that
        A4: v in Cent Q & w = x * v by Th52;
        ex u st u in Cent Q & w = y * u
        proof
          take (z \ v);
          z in [#] (lp (Cent Q)) & v in [#] (lp (Cent Q)) by A2,A4,Th25;
          then z \ v in [#] (lp (Cent Q)) by Th39;
          hence (z \ v) in Cent Q by Th25;
          w = x * (z * (z \ v)) by A4
          .= y * (z \ v) by A2,Def23,A3;
          hence thesis;
        end;
        hence thesis by Th52;
      end;
      assume w in y * lp (Cent Q);
      then consider v such that
      A5: v in Cent Q & w = y * v by Th52;
      ex u st u in Cent Q & w = x * u
      proof
        take (z * v);
        z in [#] (lp (Cent Q)) & v in [#] (lp (Cent Q)) by A2,A5,Th25;
        then z * v in [#] (lp (Cent Q)) by Th37;
        hence thesis by Def23,A3,A2,A5,Th25;
      end;
      hence thesis by Th52;
    end;
    hence x * lp (Cent Q) = y * lp (Cent Q) by SUBSET_1:3;
  end;
end;

theorem Th54:
  lp (Cent Q) is normal
proof
  set H = lp (Cent Q);
  A1: for x,y holds (x * H) * (y * H) = (x * y) * H
  proof
    let x,y;
    for z holds z in (x * H) * (y * H) iff z in (x * y) * H
    proof
      let z;
      thus z in (x * H) * (y * H) implies z in (x * y) * H
      proof
        assume z in (x * H) * (y * H);
        then consider v,w such that
        A2: v in x * H & w in y * H & z = v * w by Def42;
        consider v1 being Element of Q such that
        A3: v1 in Cent Q & v = x * v1 by Th52,A2;
        consider w1 being Element of Q such that
        A4: w1 in Cent Q & w = y * w1 by Th52,A2;
        v1 in [#] lp (Cent Q) & w1 in [#] lp (Cent Q) by A3,A4,Th25;
        then v1 * w1 in [#] lp (Cent Q) by Th37;
        then A5: v1 * w1 in Cent Q by Th25;
        A6: v1 in Comm Q by A3,XBOOLE_0:def 4;
        A7: v1 in Nucl Q by A3,XBOOLE_0:def 4;
        A8: v1 in Nucl_m Q & v1 in Nucl_r Q by A7,Th12;
        w1 in Nucl Q by A4,XBOOLE_0:def 4;
        then A9: w1 in Nucl_r Q by Th12;
        z  = ((x * v1) * y) * w1 by Def24,A9,A4,A3,A2
        .= (x * (v1 * y)) * w1 by Def23,A8
        .= (x * (y * v1)) * w1 by Def25,A6
        .= ((x * y) * v1) * w1 by Def24,A8
        .= (x * y) * (v1 * w1) by Def24,A9;
        hence z in (x * y) * H by Th52,A5;
      end;
      assume z in (x * y) * H;
      then consider w such that
      A10: w in Cent Q & z = (x * y) * w by Th52;
      w in Nucl Q by A10,XBOOLE_0:def 4;
      then A11: w in Nucl_r Q by Th12;
      ex u,v st u in x * H & v in y * H & z = u * v
      proof
        take x * 1.Q,y * w;
        thus thesis by Def24,A11,Th52, Th50,A10;
      end;
      hence z in (x * H) * (y * H) by Def42;
    end;
    hence thesis by SUBSET_1:3;
  end;
  for x,y holds (x * H) * (y * H) = (x * y) * H &
  for z holds ((x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H))
  &
  ((y * H) * (x * H) = (z * H) * (x * H) implies (y * H) = (z * H))
  proof
    let x,y;
    thus (x * H) * (y * H) = (x * y) * H by A1;
    let z;
    thus (x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H)
    proof
      assume (x * H) * (y * H) = (x * H) * (z * H);
      then (x * y) * H = (x * H) * (z * H) by A1;
      then (x * y) * H = (x * z) * H by A1;
      then consider w such that
      A12: w in Cent Q & x * z = (x * y) * w by Th53;
      w in Nucl Q by A12,XBOOLE_0:def 4;
      then A13: w in Nucl_r Q by Th12;
      x * z = x * (y * w) by A12,Def24,A13;
      hence y * H = z * H by Th1,Th53,A12;
    end;
    assume (y * H) * (x * H) = (z * H) * (x * H);
    then (y * x) * H = (z * H) * (x * H) by A1;
    then (y * x) * H = (z * x) * H by A1;
    then consider w such that
    A14: w in Cent Q & z * x = (y * x) * w by Th53;
    A15: w in Comm Q by A14,XBOOLE_0:def 4;
    w in Nucl Q by A14,XBOOLE_0:def 4;
    then A16: w in Nucl_l Q by Th12;
    z * x = w * (y * x) by A14,Def25,A15
    .= (w * y) * x by Def22,A16;
    then z = w * y by Th2;
    then z = y * w by Def25,A15;
    hence y * H = z * H by Th53,A14;
  end;
  hence thesis;
end;

begin :: AIM Conjecture

::We define the set InnAut of inner mappings of Q,
::define the notion of an AIM loop and relate this to
::the conditions on T, L, and R defined by satisfies_TT, etc.
::For AIM loops we will prove the nucleus and the center are normal.

definition
  let Q be multLoop;
  func InnAut Q -> Subset of Funcs(Q,Q) means
  :Def49:
  for f being object holds
  f in it iff
  ex g being Function of Q,Q st
  f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q;
  existence
  proof
    set I = {g where g is Function of Q,Q : g in Mlt ([#] Q) & g.(1.Q) = 1.Q};
    I c= Funcs(Q,Q)
    proof
      let f be object;
      assume f in I;
      then ex g being Function of Q,Q st
        f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q;
      hence thesis;
    end;
    then reconsider I as Subset of Funcs(Q,Q);
    take I;
    thus thesis;
  end;
  uniqueness
  proof
    let I1,I2 be Subset of Funcs(Q,Q);
    assume that
    A8: for f being object holds f in I1 iff
    ex g being Function of Q,Q st
    f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q
    and
    A9: for f being object holds f in I2 iff
    ex g being Function of Q,Q st
    f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q;
    for f being object holds f in I1 iff f in I2
    proof
      let f be object;
      f in I1 iff
      ex g being Function of Q,Q st
      f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q by A8;
      hence thesis by A9;
    end;
    hence thesis by TARSKI:2;
  end;
end;

registration
  let Q be multLoop;
  cluster InnAut Q -> non empty composition-closed inverse-closed;
  coherence
  proof
    set I = InnAut Q;
    thus A1: I is non empty
    proof
      set g = (curry (the multF of Q)).(1.Q);
      ex h being Function of Q,Q st
      g = h & h in Mlt ([#] Q) & h.(1.Q) = 1.Q
      proof
        take g;
        g.(1.Q) = 1.Q * 1.Q by FUNCT_5:69
        .= 1.Q;
        hence thesis by Th32;
      end;
      hence thesis by Def49;
    end;
    thus I is composition-closed
      proof
        let f,g be Element of I;
        consider f2 being Function of Q,Q such that
        A5: f = f2 & f2 in Mlt ([#] Q) & f2.(1.Q) = 1.Q by A1,Def49;
        consider g2 being Function of Q,Q such that
        A6: g = g2 & g2 in Mlt ([#] Q) & g2.(1.Q) = 1.Q by A1,Def49;
        set h= f2*g2;
        f*g = h & h in Mlt ([#] Q) & h.(1.Q) = 1.Q
        by A5,A6,FUNCT_2:15,Def34;
        hence thesis by Def49;
      end;
    thus I is inverse-closed
      proof
        let f be Element of I;
        consider f2 being Function of Q,Q such that
        A7: f = f2 & f2 in Mlt ([#] Q) & f2.(1.Q) = 1.Q by A1,Def49;
        ex h being Function of Q,Q st
        f" = h & h in Mlt ([#] Q) & h.(1.Q) = 1.Q
        proof
          reconsider f2 as Permutation of the carrier of Q by Th35,A7;
          take f2";
          (f2").(1.Q) = ((f2") * f2).(1.Q) by FUNCT_2:15,A7
          .= (id the carrier of Q).(1.Q) by FUNCT_2:61
          .= 1.Q;
          hence thesis by A7,Def35;
        end;
        hence thesis by Def49;
      end;
    end;
  end;

theorem Th55:
  for f being Function of Q,Q holds
  f in InnAut Q iff f in Mlt ([#] Q) & f.(1.Q) = 1.Q
proof
  let f be Function of Q,Q;
  thus f in InnAut Q implies f in Mlt ([#] Q) & f.(1.Q) = 1.Q
  proof
    assume f in InnAut Q;
    then ex g being Function of Q,Q st
    f = g & g in Mlt ([#] Q) & g.(1.Q) = 1.Q by Def49;
    hence thesis;
  end;
  thus thesis by Def49;
end;

definition
  let Q be multLoop;
  attr Q is AIM means :Def50:
    for f,g being Function of Q,Q st f in InnAut Q & g in InnAut Q holds
      f*g = g*f;
end;

definition
  let Q,x; 
  deffunc Tx(Element of Q) = T_map($1,x);
  func T_MAP(x) -> Function of Q,Q means :TM1:
  for u holds it.u = T_map(u,x);
  existence
  proof
    ex f being Function of Q,Q st
    for u being Element of Q holds f.u = Tx(u) from FUNCT_2:sch 4;
    hence thesis;
  end;
  uniqueness
  proof
    let f,g be Function of Q,Q such that
A1: for u holds f.u = Tx(u) and
A2: for u holds g.u = Tx(u);
    let u;
    thus f.u = Tx(u) by A1
    .= g.u by A2;
  end;
end;

theorem Th56:
  T_MAP(x) in InnAut Q
proof
  set f = T_MAP(x);
  reconsider g = (curry (the multF of Q)).x as
    Permutation of the carrier of Q by Th30;
  reconsider h = (curry' (the multF of Q)).x as
    Permutation of the carrier of Q by Th31;
  A2: f = g" * h
  proof
    for u holds (g * f).u = h.u
    proof
      let u;
      thus (g * f).u = g.(f.u) by FUNCT_2:15
      .= g.(T_map(u,x)) by TM1
      .= x * (x \ (u * x)) by FUNCT_5:69
      .= h.u by FUNCT_5:70;
    end;
    then g"*h = g"*(g*f) by FUNCT_2:def 8
    .= (g"*g)*f by RELAT_1:36
    .= (id the carrier of Q)*f by FUNCT_2:61
    .= f by FUNCT_2:17;
    hence thesis;
  end;
  g in Mlt ([#] Q) by Th32;
  then A3: g" in Mlt ([#] Q) by Def35;
  A4: h in Mlt ([#] Q) by Th33;
  f.(1.Q) = T_map(1.Q,x) by TM1
  .= 1.Q by Th5;
  hence thesis by A4,Th55,A2,Def34,A3;
end;

definition
  let Q,x,y;
  deffunc Lx(Element of Q) = L_map($1,x,y);
  func L_MAP(x,y) -> Function of Q,Q means :LM1:
  for u holds it.u = L_map(u,x,y);
  existence
  proof
    ex f being Function of Q,Q st
    for u being Element of Q holds f.u = Lx(u) from FUNCT_2:sch 4;
    hence thesis;
  end;
  uniqueness
  proof
    let f,g be Function of Q,Q such that
A1: for u holds f.u = Lx(u) and
A2: for u holds g.u = Lx(u);
    let u;
    thus f.u = Lx(u) by A1
    .= g.u by A2;
  end;
end;

theorem Th57:
  L_MAP(x,y) in InnAut Q
proof
  set f = L_MAP(x,y);
  reconsider g = (curry (the multF of Q)).(y * x) as
    Permutation of the carrier of Q by Th30;
  reconsider h = (curry (the multF of Q)).x as
    Permutation of the carrier of Q by Th30;
  reconsider k = (curry (the multF of Q)).(y) as
    Permutation of the carrier of Q by Th30;
  A2: f = g" * (k * h)
  proof
    for u holds (g*f).u = (k*h).u
    proof
      let u;
      (g*f).u = g.(f.u) by FUNCT_2:15
      .= g.(L_map(u,x,y)) by LM1
      .= (y * x) * ((y * x) \ (y * (x * u))) by FUNCT_5:69
      .= k.(x * u) by FUNCT_5:69
      .= k.(h.u) by FUNCT_5:69
      .= (k*h).u by FUNCT_2:15;
      hence thesis;
    end;
    then g"*(k*h) = g"*(g*f) by FUNCT_2:def 8
    .= (g"*g)*f by RELAT_1:36
    .= (id the carrier of Q)*f by FUNCT_2:61
    .= f by FUNCT_2:17;
    hence thesis;
  end;
  g in Mlt ([#] Q) by Th32;
  then A3: g" in Mlt ([#] Q) by Def35;
  h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th32;
  then A4:k * h in Mlt ([#] Q) by Def34;
  f.(1.Q) = L_map(1.Q,x,y) by LM1
  .= 1.Q by Th5;
  hence thesis by Th55,A4,A2,Def34,A3;
end;

definition
  let Q,x,y;
  deffunc Rx(Element of Q) = R_map($1,x,y);
  func R_MAP(x,y) -> Function of Q,Q means :RM1:
  for u holds it.u = R_map(u,x,y);
  existence
  proof
    ex f being Function of Q,Q st
    for u being Element of Q holds f.u = Rx(u) from FUNCT_2:sch 4;
    hence thesis;
  end;
  uniqueness
  proof
    let f,g be Function of Q,Q such that
A1: for u holds f.u = Rx(u) and
A2: for u holds g.u = Rx(u);
    let u;
    thus f.u = Rx(u) by A1
    .= g.u by A2;
  end;
end;

theorem Th58:
  R_MAP(x,y) in InnAut Q
proof
  set f = R_MAP(x,y);
  reconsider g = (curry' (the multF of Q)).(x * y) as
    Permutation of the carrier of Q by Th31;
  reconsider h = (curry' (the multF of Q)).x as
    Permutation of the carrier of Q by Th31;
  reconsider k = (curry' (the multF of Q)).(y) as
    Permutation of the carrier of Q by Th31;
  A2: f = g" * (k * h)
  proof
    for u holds (g*f).u = (k*h).u
    proof
      let u;
      thus (g*f).u = g.(f.u) by FUNCT_2:15
      .= g.(R_map(u,x,y)) by RM1
      .= (((u * x) * y) / (x * y)) * (x * y) by FUNCT_5:70
      .= k.(u * x) by FUNCT_5:70
      .= k.(h.u) by FUNCT_5:70
      .= (k*h).u by FUNCT_2:15;
    end;
    then g"*(k*h) = g"*(g*f) by FUNCT_2:def 8
    .= (g"*g)*f by RELAT_1:36
    .= (id Q)*f by FUNCT_2:61
    .= f by FUNCT_2:17;
    hence thesis;
  end;
  g in Mlt ([#] Q) by Th33;
  then A3: g" in Mlt ([#] Q) by Def35;
  h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th33;
  then A4: k * h in Mlt ([#] Q) by Def34;
  f.(1.Q) = R_map(1.Q,x,y) by RM1
  .= 1.Q by Th6;
  hence thesis by Th55,A4,A2,Def34,A3;
end;

registration
  cluster Trivial-multLoopStr -> AIM;
  coherence
  proof
    set Q = Trivial-multLoopStr;
    let f,g be Function of Q,Q;
    for x being Element of Q holds (f*g).x = (g*f).x by ALGSTR_1:9;
    hence thesis by FUNCT_2:def 8;
  end;
end;

registration
  cluster non empty strict AIM for multLoop;
  existence
  proof
    take Trivial-multLoopStr;
    thus thesis;
  end;
end;

registration 
  cluster -> satisfying_TT satisfying_TL satisfying_TR satisfying_LR
             satisfying_LL satisfying_RR for AIM multLoop;
coherence
proof
  let Q be AIM multLoop;
  thus Q is satisfying_TT
  proof
    let u,x,y be Element of Q;
    set f = T_MAP(x);
    set g = T_MAP(y);
    A3: f in InnAut Q & g in InnAut Q by Th56;
    T_map(T_map(u,x),y) = T_map(f.u,y) by TM1
    .= g.(f.u) by TM1
    .= (g*f).u by FUNCT_2:15
    .= (f*g).u by A3,Def50
    .= f.(g.u) by FUNCT_2:15
    .= T_map(g.u,x) by TM1
    .= T_map(T_map(u,y),x) by TM1;
    hence thesis;
  end;
  thus Q is satisfying_TL
  proof
    let u,x,y,z be Element of Q;
    set f = L_MAP(x,y);
    set g = T_MAP(z);
    A6: f in InnAut Q & g in InnAut Q by Th56,Th57;
    T_map(L_map(u,x,y),z) = T_map(f.u,z) by LM1
    .= g.(f.u) by TM1
    .= (g*f).u by FUNCT_2:15
    .= (f*g).u by A6,Def50
    .= f.(g.u) by FUNCT_2:15
    .= L_map(g.u,x,y) by LM1
    .= L_map(T_map(u,z),x,y) by TM1;
    hence thesis;
  end;
  thus Q is satisfying_TR
  proof
    let u,x,y,z be Element of Q;
    set f = R_MAP(x,y);
    set g = T_MAP(z);
    A9: f in InnAut Q & g in InnAut Q by Th56,Th58;
    T_map(R_map(u,x,y),z) = T_map(f.u,z) by RM1
    .= g.(f.u) by TM1
    .= (g*f).u by FUNCT_2:15
    .= (f*g).u by A9,Def50
    .= f.(g.u) by FUNCT_2:15
    .= R_map(g.u,x,y) by RM1
    .= R_map(T_map(u,z),x,y) by TM1;
    hence thesis;
  end;
  thus Q is satisfying_LR
  proof
    let u,x,y,z,w be Element of Q;
    set f = R_MAP(x,y);
    set g = L_MAP(z,w);
    A12: f in InnAut Q & g in InnAut Q by Th58,Th57;
    L_map(R_map(u,x,y),z,w) = L_map(f.u,z,w) by RM1
    .= g.(f.u) by LM1
    .= (g*f).u by FUNCT_2:15
    .= (f*g).u by A12,Def50
    .= f.(g.u) by FUNCT_2:15
    .= R_map(g.u,x,y) by RM1
    .= R_map(L_map(u,z,w),x,y) by LM1;
    hence thesis;
  end;
  thus Q is satisfying_LL
  proof
    let u,x,y,z,w be Element of Q;
    set f = L_MAP(x,y);
    set g = L_MAP(z,w);
    A15: f in InnAut Q & g in InnAut Q by Th57;
    L_map(L_map(u,x,y),z,w) = L_map(f.u,z,w) by LM1
    .= g.(f.u) by LM1
    .= (g*f).u by FUNCT_2:15
    .= (f*g).u by A15,Def50
    .= f.(g.u) by FUNCT_2:15
    .= L_map(g.u,x,y) by LM1
    .= L_map(L_map(u,z,w),x,y) by LM1;
    hence thesis;
  end;
  let u,x,y,z,w be Element of Q;
  set f = R_MAP(x,y);
  set g = R_MAP(z,w);
  A18:f in InnAut Q & g in InnAut Q by Th58;
  R_map(R_map(u,x,y),z,w) = R_map(f.u,z,w) by RM1
  .= g.(f.u) by RM1
  .= (g*f).u by FUNCT_2:15
  .= (f*g).u by A18,Def50
  .= f.(g.u) by FUNCT_2:15
  .= R_map(g.u,x,y) by RM1
  .= R_map(R_map(u,z,w),x,y) by RM1;
  hence thesis;
end;
end;

theorem Th59:
  for f being Function of Q,Q st f in Mlt (Nucl Q) holds
  ex u,v st u in Nucl Q & v in Nucl Q & for x holds f.x = u * (x * v)
proof
  set H = Nucl Q;
  defpred P[Function of Q,Q] means
  ex u,v st u in Nucl Q & v in Nucl Q & for x holds $1.x = u * (x * v);
  A1: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds f.x = x * u
    holds P[f]
  proof
    let u such that  A2: u in H;
    let f be Function of Q,Q such that A3: for x holds f.x = x * u;
    take 1.Q,u;
    thus thesis by A3,A2,Th20;
  end;
  A4: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds f.x = u * x
    holds P[f]
  proof
    let u such that A5: u in H;
    let f be Function of Q,Q such that A6: for x holds f.x = u * x;
    take u, 1.Q;
    thus thesis by A6,A5,Th20;
  end;
  A7: for g,h being Permutation of the carrier of Q st P[g] & P[h] holds P[g*h]
  proof
    let g,h be Permutation of the carrier of Q;
    assume A8: P[g] & P[h];
    consider u,v such that
    A9: u in H & v in H & for x holds g.x = u * (x * v) by A8;
    consider z,w such that
    A10: z in H & w in H & for x holds h.x = z * (x * w) by A8;
    take u * z, w * v;
    u in [#] (lp (Nucl Q)) & z in [#] (lp (Nucl Q)) by Th24,A9,A10;
    then u * z in [#] (lp (Nucl Q)) by Th37;
    hence u * z in H by Th24;
    w in [#] (lp (Nucl Q)) & v in [#] (lp (Nucl Q)) by Th24,A9,A10;
    then A11: w * v in [#] (lp (Nucl Q)) by Th37;
    then A12: w * v in Nucl Q by Th24;
    thus w * v in H by A11,Th24;
    A13: u in Nucl_l Q by A9,Th12;
    A14: v in Nucl_r Q by A9,Th12;
    A15: w in Nucl_r Q by A10,Th12;
    A16: w * v in Nucl_r Q by A12,Th12;
    let x;
    (g*h).x = g.(h.x) by FUNCT_2:15
    .= g.(z * (x * w)) by A10
    .= u * ((z * (x * w)) * v) by A9
    .= (u * (z * (x * w))) * v by A13,Def22
    .= ((u * z) * (x * w)) * v by A13,Def22
    .= (((u * z) * x) * w) * v by A15,Def24
    .= ((u * z) * x) * (w * v) by A14,Def24
    .= (u * z) * (x * (w * v)) by A16,Def24;
    hence thesis;
  end;
  A17: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of Q;
    assume P[g];
    then consider u,v such that
    A18: u in H & v in H & for x holds g.x = u * (x * v);
    A19: u in Nucl_m Q by A18,Th12;
    A20: v in Nucl_m Q & v in Nucl_r Q by A18,Th12;
    take 1.Q / u,v \ 1.Q;
    1.Q in Nucl Q by Th20;
    then A21: 1.Q in [#] (lp (Nucl Q)) by Th24;
    u in [#] (lp (Nucl Q)) by Th24,A18;
    then 1.Q / u in [#] (lp (Nucl Q)) by Th41,A21;
    hence 1.Q / u in Nucl Q by Th24;
    v in [#] (lp (Nucl Q)) by Th24,A18;
    then v \ 1.Q in [#] (lp (Nucl Q)) by Th39,A21;
    hence v \ 1.Q in Nucl Q by Th24;
    let x;
    reconsider k = (curry (the multF of Q)).(1.Q / u) as Permutation of Q
    by Th30;
    reconsider h = (curry' (the multF of Q)).(v \ 1.Q) as Permutation of Q
    by Th31;
    (k*h)*g = id Q
    proof
      for y holds ((k*h)*g).y = (id Q).y
      proof
        let y;
        ((k*h)*g).y = (k*h).(g.y) by FUNCT_2:15
        .= (k*h).(u * (y * v)) by A18
        .= k.(h.(u * (y * v))) by FUNCT_2:15
        .= k.((u * (y * v)) * (v \ 1.Q)) by FUNCT_5:70
        .= k.(((u * y) * v) * (v \ 1.Q)) by Def24,A20
        .= k.((u * y) * (v * (v \ 1.Q))) by Def23,A20
        .= (1.Q / u) * (u * y) by FUNCT_5:69
        .= ((1.Q / u) * u) * y by Def23,A19
        .= y;
        hence thesis;
      end;
      hence thesis by FUNCT_2:def 8;
    end;
    then (g").x = (k*h).x by FUNCT_2:60
    .= k.(h.x) by FUNCT_2:15
    .= k.(x * (v \ 1.Q)) by FUNCT_5:70
    .= (1.Q / u) * (x * (v \ 1.Q)) by FUNCT_5:69;
    hence thesis;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A4,A7,A17);
  hence thesis;
end;

theorem Th60:
  y in x * lp (Nucl Q) iff
  ex u,v st u in Nucl Q & v in Nucl Q & y = u * (x * v)
proof
  thus y in x * lp (Nucl Q) implies ex u,v st u in Nucl Q & v in Nucl Q &
    y = u * (x * v)
  proof
    assume y in x * lp (Nucl Q);
    then y in x * Nucl Q by Th24;
    then consider h being Permutation of the carrier of Q such that
    A1: h in Mlt (Nucl Q) & h.x = y by Def39;
    consider u,v such that
    A2: u in Nucl Q & v in Nucl Q & for z holds h.z = u * (z * v)
      by Th59,A1;
    take u,v;
    thus thesis by A1,A2;
  end;
  given u,v such that
  A3: u in Nucl Q & v in Nucl Q & y = u * (x * v);
  ex h being Permutation of the carrier of Q st h in Mlt (Nucl Q) & h.x = y
  proof
    reconsider h = (curry' (the multF of Q)).(v),
    k = (curry (the multF of Q)).u
    as Permutation of the carrier of Q
    by Th31,Th30;
    take k*h;
    h in Mlt (Nucl Q) & k in Mlt (Nucl Q) by Th33,Th32,A3;
    hence k*h in Mlt (Nucl Q) by Def34;
    (k*h).x = k.(h.x) by FUNCT_2:15
    .= k.(x * v) by FUNCT_5:70
    .= y by A3, FUNCT_5:69;
    hence thesis;
  end;
  then y in x * Nucl Q by Def39;
  hence thesis by Th24;
end;

theorem Th61:
  x * lp (Nucl Q) = y * lp (Nucl Q) iff
    ex u,v st u in Nucl Q & v in Nucl Q & y = u * (x * v)
proof
  thus x * lp (Nucl Q) = y * lp (Nucl Q) implies
    ex u,v st u in Nucl Q & v in Nucl Q & y = u * (x * v)
  proof
    assume A1: x * lp (Nucl Q) = y * lp (Nucl Q);
    A2: 1.Q in Nucl Q by Th20;
    y = 1.Q * (y * 1.Q);
    hence ex u,v st u in Nucl Q & v in Nucl Q & y = u * (x * v)
      by Th60,A1,A2;
  end;
  given u,v such that
  A3: u in Nucl Q & v in Nucl Q & y = u * (x * v);
  A4: u in Nucl_l Q & u in Nucl_m Q by A3,Th12;
  A5: v in Nucl_m Q & v in Nucl_r Q by A3,Th12;
  for w holds w in x * lp (Nucl Q) iff w in y * lp (Nucl Q)
  proof
    let w;
    thus w in x * lp (Nucl Q) implies w in y * lp (Nucl Q)
    proof
      assume w in x * lp (Nucl Q);
      then consider u1,v1 being Element of Q such that
      A6: u1 in Nucl Q & v1 in Nucl Q & w = u1 * (x * v1) by Th60;
      ex u2,v2 being Element of Q st u2 in Nucl Q & v2 in Nucl Q &
      w = u2 * (y * v2)
      proof
        take u1 / u,v \ v1;
        u in [#] (lp (Nucl Q)) & u1 in [#] (lp (Nucl Q))
        by A3,A6,Th24;
        then u1 / u in [#] (lp (Nucl Q)) by Th41;
        hence (u1 / u) in Nucl Q by Th24;
        v in [#] (lp (Nucl Q)) & v1 in [#] (lp (Nucl Q))
        by A3,A6,Th24;
        then v \ v1 in [#] (lp (Nucl Q)) by Th39;
        hence (v \ v1) in Nucl Q by Th24;
        w  = u1 * (x * (v * (v \ v1))) by A6
        .= ((u1 / u) * u) * ((x * v) * (v \ v1)) by Def23,A5
        .= (u1 / u) * (u * ((x * v) * (v \ v1))) by Def23,A4
        .= (u1 / u) * (y * (v \ v1)) by A3,Def22,A4;
        hence thesis;
      end;
      hence thesis by Th60;
    end;
    thus w in y * lp (Nucl Q) implies w in x * lp (Nucl Q)
    proof
      assume w in y * lp (Nucl Q);
      then consider u1,v1 being Element of Q such that
      A7: u1 in Nucl Q & v1 in Nucl Q & w = u1 * (y * v1) by Th60;
      ex u2,v2 being Element of Q st u2 in Nucl Q & v2 in Nucl Q &
      w = u2 * (x * v2)
      proof
        take u1 * u,v * v1;
        u in [#] (lp (Nucl Q)) & u1 in [#] (lp (Nucl Q))
        by A3,A7,Th24;
        then u1 * u in [#] (lp (Nucl Q)) by Th37;
        hence (u1 * u) in Nucl Q by Th24;
        v in [#] (lp (Nucl Q)) & v1 in [#] (lp (Nucl Q))
        by A3,A7,Th24;
        then v * v1 in [#] (lp (Nucl Q)) by Th37;
        hence (v * v1) in Nucl Q by Th24;
        w = u1 * (((u * x) * v) * v1) by A3,A7,Def24,A5
        .= u1 * ((u * x) * (v * v1)) by Def23,A5
        .= u1 * (u * (x * (v * v1))) by Def22,A4
        .= (u1 * u) * (x * (v * v1)) by Def23,A4;
        hence thesis;
      end;
      hence thesis by Th60;
    end;
  end;
  hence x * lp (Nucl Q) = y * lp (Nucl Q) by SUBSET_1:3;
end;

:: Suggested result and proof by Kinyon Sep 10 2018
:: as crucial part of proving the nucleus of an AIM loop
:: is normal.
theorem Th62:
  for Q being AIM multLoop holds
  for x,u being Element of Q holds
  u in Nucl Q implies T_map(u,x) in Nucl Q
proof
  let Q be AIM multLoop;
  let x,u be Element of Q;
  assume u in Nucl Q;
  then A1: u in Nucl_l Q & u in Nucl_m Q & u in Nucl_r Q by Th12;
  for y,z being Element of Q holds (T_map(u,x) * y) * z = T_map(u,x) * (y * z)
  proof
    let y,z be Element of Q;
    Q is satisfying_TR;
    then R_map(T_map(u,x),y,z) = T_map(R_map(u,y,z),x)
    .= T_map((u * (y * z)) / (y * z),x) by Def22,A1
    .= T_map(u,x);
    hence thesis;
  end;
  then A2: T_map(u,x) in Nucl_l Q by Def22;
  for y,z being Element of Q holds (y * z) * T_map(u,x) = y * (z * T_map(u,x))
  proof
    let y,z be Element of Q;
    Q is satisfying_TL;
    then L_map(T_map(u,x),z,y) = T_map(L_map(u,z,y),x)
    .= T_map((y * z) \ ((y * z) * u),x) by Def24,A1
    .= T_map(u,x);
    hence thesis;
  end;
  then A3: T_map(u,x) in Nucl_r Q by Def24;
  for y,z being Element of Q holds (y * T_map(u,x)) * z = y * (T_map(u,x) * z)
  proof
    let y,z be Element of Q;
    deffunc M(Element of Q) = y \ ((y * ($1 * z)) / z);
    A4: M(u) = y \ (((y * u) * z) / z) by Def23,A1
    .= u;
    consider m be Function of Q,Q
    such that
    A5: for v being Element of Q holds m.v = M(v)
    from FUNCT_2:sch 4;
    A6: m in InnAut Q
    proof
      reconsider h = (curry' (the multF of Q)).z,
               k = (curry (the multF of Q)).y as Permutation of Q
        by Th31,Th30;
      A7: h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th32,Th33;
      then A8: h" in Mlt ([#] Q) & k" in Mlt ([#] Q) by Def35;
      k*h in Mlt ([#] Q) by A7,Def34;
      then h"*(k*h) in Mlt ([#] Q) by A8,Def34;
      then A9: k"*(h"*(k*h)) in Mlt ([#] Q) by A8,Def34;
      A10: for v being Element of Q holds  (h*k).v = (y * v) * z
      proof
        let v be Element of Q;
        (h*k).v = h.(k.v) by FUNCT_2:15
        .= h.(y * v) by FUNCT_5:69
        .= (y * v) * z by FUNCT_5:70;
        hence thesis;
      end;
      A11: for v being Element of Q holds  (k*h).v = y * (v * z)
      proof
        let v be Element of Q;
        (k*h).v = k.(h.v) by FUNCT_2:15
        .= k.(v * z) by FUNCT_5:70
        .= y * (v * z) by FUNCT_5:69;
        hence thesis;
      end;
      for v being Element of Q holds m.v = (k"*(h"*(k*h))).v
      proof
        let v be Element of Q;
        (y * (m.v)) * z = (y * M(v)) * z by A5
        .= (k*h).v by A11
        .= ((id the carrier of Q)*(k*h)).v by FUNCT_2:17
        .= ((h*h")*(k*h)).v by FUNCT_2:61
        .= (h*(h"*(k*h))).v by RELAT_1:36
        .= (h*((id Q)*(h"*(k*h)))).v by FUNCT_2:17
        .= (h*((k*k")*(h"*(k*h)))).v by FUNCT_2:61
        .= (h*(k*(k"*(h"*(k*h))))).v by RELAT_1:36
        .= ((h*k)*(k"*(h"*(k*h)))).v by RELAT_1:36
        .= (h*k).((k"*(h"*(k*h))).v) by FUNCT_2:15
        .= (y * ((k"*(h"*(k*h))).v)) * z by A10;
        then y * (m.v) = y * ((k"*(h"*(k*h))).v) by Th2;
        hence thesis by Th1;
      end;
      then  A12: m in Mlt ([#] Q) by A9,FUNCT_2:def 8;
      m.(1.Q) = M(1.Q) by A5
      .= 1.Q by Th5;
      hence thesis by Def49,A12;
    end;
    set t = T_MAP(x);
    t in InnAut Q by Th56;
    then A14: t*m = m*t by Def50,A6;
    M(T_map(u,x)) = m.(T_map(u,x)) by A5
    .= m.(t.u) by TM1
    .= (m*t).u by FUNCT_2:15
    .= t.(m.u) by FUNCT_2:15,A14
    .= t.(M(u)) by A5
    .= T_map(u,x) by A4,TM1;
    hence thesis;
  end;
  then T_map(u,x) in Nucl_m Q by Def23;
  hence thesis by Th12,A2,A3;
end;

theorem Th63:
  for Q being AIM multLoop holds
  for x,u being Element of Q holds
    u in Nucl Q implies (x * u) / x in Nucl Q
proof
  let Q be AIM multLoop, x,u be Element of Q;
  assume u in Nucl Q;
  then A1: u in Nucl_l Q & u in Nucl_m Q & u in Nucl_r Q by Th12;
  deffunc Tdx(Element of Q) = (x * $1) / x;
  consider t be Function of Q,Q
  such that
  A2: for v being Element of Q holds t.v = Tdx(v)
  from FUNCT_2:sch 4;
  A3: t in InnAut Q
  proof
    reconsider g = (curry (the multF of Q)).x,
               h = (curry' (the multF of Q)).x as Permutation of Q
      by Th30,Th31;
    A4: t = h" * g
    proof
      for u being Element of Q holds (h * t).u = g.u
      proof
        let u be Element of Q;
        (h * t).u = h.(t.u) by FUNCT_2:15
        .= h.(Tdx(u)) by A2
        .= ((x * u) / x) * x by FUNCT_5:70
        .= g.u by FUNCT_5:69;
        hence thesis;
      end;
      then h"*g = h"*(h*t) by FUNCT_2:def 8
      .= (h"*h)*t by RELAT_1:36
      .= (id the carrier of Q)*t by FUNCT_2:61
      .= t by FUNCT_2:17;
      hence thesis;
    end;
    A5: g in Mlt ([#] Q) by Th32;
    h in Mlt ([#] Q) by Th33;
    then A6: h" in Mlt ([#] Q) by Def35;
    t.(1.Q) = (x * 1.Q) / x by A2
    .= 1.Q by Th6;
    hence thesis by Th55,A6,A4,Def34,A5;
  end;
  for y,z being Element of Q holds (Tdx(u) * y) * z = Tdx(u) * (y * z)
  proof
    let y,z be Element of Q;
    set f = R_MAP(y,z);
    A8: f in InnAut Q by Th58;
    f.u = R_map(u,y,z) by RM1
    .= (u * (y * z)) / (y * z) by Def22,A1
    .= u;
    then Tdx(u) = t.(f.u) by A2
    .= (t*f).u by FUNCT_2:15
    .= (f*t).u by A8,Def50,A3
    .= f.(t.u) by FUNCT_2:15
    .= f.(Tdx(u)) by A2
    .= R_map(Tdx(u),y,z) by RM1
    .= ((Tdx(u) * y) * z) / (y * z);
    hence thesis;
  end;
  then A9: Tdx(u) in Nucl_l Q by Def22;
  for y,z being Element of Q holds (y * z) * Tdx(u) = y * (z * Tdx(u))
  proof
    let y,z be Element of Q;
    set f = L_MAP(z,y);
    f in InnAut Q by Th57;
    then A11: t*f = f*t by Def50,A3;
    f.u = L_map(u,z,y) by LM1
    .= (y * z) \ ((y * z) * u) by Def24,A1
    .= u;
    then Tdx(u) = t.(f.u) by A2
    .= (t*f).u by FUNCT_2:15
    .= f.(t.u) by FUNCT_2:15,A11
    .= f.(Tdx(u)) by A2
    .= L_map(Tdx(u),z,y) by LM1
    .= (y * z) \ (y * (z * Tdx(u)));
    hence thesis;
  end;
  then A12: Tdx(u) in Nucl_r Q by Def24;
  for y,z being Element of Q holds (y * Tdx(u)) * z = y * (Tdx(u) * z)
  proof
    let y,z be Element of Q;
    deffunc M(Element of Q) = y \ ((y * ($1 * z)) / z);
    A13: M(u) = y \ (((y * u) * z) / z) by Def23,A1
    .= u;
    consider m be Function of Q,Q such that
    A14: for v being Element of Q holds m.v = M(v) from FUNCT_2:sch 4;
    A15: m in InnAut Q
    proof
      reconsider h = (curry' (the multF of Q)).(z),
                 k = (curry (the multF of Q)).(y) as Permutation of Q
        by Th31,Th30;
      A16: h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th32,Th33;
      then A17: h" in Mlt ([#] Q) & k" in Mlt ([#] Q) by Def35;
      k*h in Mlt ([#] Q) by A16,Def34;
      then h"*(k*h) in Mlt ([#] Q) by A17,Def34;
      then A18: k"*(h"*(k*h)) in Mlt ([#] Q) by A17,Def34;
      A19: for v being Element of Q holds  (h*k).v = (y * v) * z
      proof
        let v be Element of Q;
        (h*k).v = h.(k.v) by FUNCT_2:15
        .= h.(y * v) by FUNCT_5:69
        .= (y * v) * z by FUNCT_5:70;
        hence thesis;
      end;
      A20: for v being Element of Q holds  (k*h).v = y * (v * z)
      proof
        let v be Element of Q;
        (k*h).v = k.(h.v) by FUNCT_2:15
        .= k.(v * z) by FUNCT_5:70
        .= y * (v * z) by FUNCT_5:69;
        hence thesis;
      end;
      for v being Element of Q holds m.v = (k"*(h"*(k*h))).v
      proof
        let v be Element of Q;
        (y * (m.v)) * z = (y * M(v)) * z by A14
        .= (k*h).v by A20
        .= ((id the carrier of Q)*(k*h)).v by FUNCT_2:17
        .= ((h*h")*(k*h)).v by FUNCT_2:61
        .= (h*(h"*(k*h))).v by RELAT_1:36
        .= (h*((id the carrier of Q)*(h"*(k*h)))).v by FUNCT_2:17
        .= (h*((k*k")*(h"*(k*h)))).v by FUNCT_2:61
        .= (h*(k*(k"*(h"*(k*h))))).v by RELAT_1:36
        .= ((h*k)*(k"*(h"*(k*h)))).v by RELAT_1:36
        .= (h*k).((k"*(h"*(k*h))).v) by FUNCT_2:15
        .= (y * ((k"*(h"*(k*h))).v)) * z by A19;
        then y * (m.v) = y * ((k"*(h"*(k*h))).v) by Th2;
        hence thesis by Th1;
      end;
      then A21: m in Mlt ([#] Q) by A18,FUNCT_2:def 8;
      m.(1.Q) = M(1.Q) by A14
      .= 1.Q by Th5;
      hence thesis by Def49,A21;
    end;
    A22: t*m = m*t by Def50,A15,A3;
    M(Tdx(u)) = m.(Tdx(u)) by A14
    .= m.(t.u) by A2
    .= (t*m).u by A22,FUNCT_2:15
    .= t.(m.u) by FUNCT_2:15
    .= t.(M(u)) by A14
    .= Tdx(u) by A13,A2;
    hence thesis;
  end;
  then Tdx(u) in Nucl_m Q by Def23;
  hence thesis by Th12,A9,A12;
end;

:: This proof was difficult and required a hint from Kinyon.
:: Kinyon's hint was essentially the proof of NuclT above.
theorem Th64:
  Q is AIM implies lp (Nucl Q) is normal 
proof
  assume A1: Q is AIM;
  set H = lp (Nucl Q);
  A2: for x,y being Element of Q holds
  (ex v being Element of Q st v in Nucl Q & y = x * v)
  iff
  (ex u,v being Element of Q st u in Nucl Q & v in Nucl Q & y = u * (x * v))
  proof
    let x,y;
    thus (ex v being Element of Q st v in Nucl Q & y = x * v)
    implies
    (ex u,v being Element of Q st u in Nucl Q & v in Nucl Q & y = u * (x * v))
    proof
      given v being Element of Q such that
      A3: v in Nucl Q & y = x * v;
      take 1.Q,v;
      thus thesis by A3,Th20;
    end;
    thus (ex u,v being Element of Q st u in Nucl Q & v in Nucl Q &
      y = u * (x * v))
    implies
    (ex v being Element of Q st v in Nucl Q & y = x * v)
    proof
      given u,v being Element of Q such that
      A4: u in Nucl Q & v in Nucl Q & y = u * (x * v);
      take T_map(u,x) * v;
      T_map(u,x) in Nucl Q by A1,Th62,A4;
      then T_map(u,x) in [#] lp (Nucl Q) & v in [#] lp (Nucl Q)
        by A4,Th24;
      then T_map(u,x) * v in [#] lp (Nucl Q) by Th37;
      hence T_map(u,x) * v in Nucl Q by Th24;
      A5: v in Nucl_r Q by Th12,A4;
      y  = (x * (x \ (u * x))) * v by Def24,A5,A4
      .= x * (T_map(u,x) * v) by Def24,A5;
      hence thesis;
    end;
  end;
  A6: for x,y being Element of Q holds y in x * H iff
  ex v being Element of Q st v in Nucl Q & y = x * v
  proof
    let x,y;
    y in x * H iff ex u,v being Element of Q st u in Nucl Q &
      v in Nucl Q & y = u * (x * v) by Th60;
    hence thesis by A2;
  end;
  A7: for x,y being Element of Q holds x * H = y * H iff
        ex v being Element of Q st v in Nucl Q & y = x * v
  proof
    let x,y;
    x * H = y * H iff ex u,v being Element of Q st u in Nucl Q & v in Nucl Q
      & y = u * (x * v) by Th61;
    hence thesis by A2;
  end;
  A8: for x,y holds (x * H) * (y * H) = (x * y) * H
  proof
    let x,y;
    for z holds z in (x * H) * (y * H) iff z in (x * y) * H
    proof
      let z;
      thus z in (x * H) * (y * H) implies z in (x * y) * H
      proof
        assume z in (x * H) * (y * H);
        then consider x2,y2 being Element of Q such that
        A9: x2 in x * H & y2 in y * H & z = x2 * y2 by Def42;
        ex v being Element of Q st v in Nucl Q & z = (x * y) * v
        proof
          consider u being Element of Q such that
          A10: u in Nucl Q & x2 = x * u by A6,A9;
          consider v being Element of Q such that
          A11: v in Nucl Q & y2 = y * v by A6,A9;
          take (T_map(u,y) * v);
          T_map(u,y) in Nucl Q by A1,Th62,A10;
          then T_map(u,y) in [#] lp (Nucl Q) & v in [#] lp (Nucl Q)
            by A11,Th24;
          then T_map(u,y) * v in [#] lp (Nucl Q) by Th37;
          hence A12: T_map(u,y) * v in Nucl Q by Th24;
          A13: u in Nucl_m Q by Th12,A10;
          A14: v in Nucl_r Q by Th12,A11;
          A15: T_map(u,y) * v in Nucl_r Q by Th12,A12;
          z = x * (u * (y * v)) by Def23,A13,A11,A10,A9
          .= x * ((y * T_map(u,y)) * v) by Def24,A14
          .= x * (y * (T_map(u,y) * v)) by Def24,A14
          .= (x * y) * (T_map(u,y) * v) by Def24,A15;
          hence thesis;
        end;
        hence z in (x * y) * H by A6;
      end;
      assume z in (x * y) * H;
      then consider v such that
      A16: v in Nucl Q & z = (x * y) * v by A6;
      ex x1,y1 being Element of Q st x1 in x * H & y1 in y * H & z = x1 * y1
      proof
        take x, y * v;
        A17:  1.Q in Nucl Q & x = x * 1.Q by Th20;
        v in Nucl_r Q by Th12,A16;
        hence thesis by A17,A16,Def24,A6;
      end;
      hence thesis by Def42;
    end;
    hence thesis by SUBSET_1:3;
  end;
  for x,y holds
  (x * H) * (y * H) = (x * y) * H
  &
  for z holds
  ((x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H))
  &
  ((y * H) * (x * H) = (z * H) * (x * H) implies (y * H) = (z * H))
  proof
    let x,y;
    thus (x * H) * (y * H) = (x * y) * H by A8;
    let z;
    thus (x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H)
    proof
      assume (x * H) * (y * H) = (x * H) * (z * H);
      then (x * y) * H = (x * H) * (z * H) by A8;
      then (x * y) * H = (x * z) * H by A8;
      then consider w such that
      A18: w in Nucl Q & x * z = (x * y) * w by A7;
      A19: w in Nucl_r Q by Th12,A18;
      x * z = x * (y * w) by A18,Def24,A19;
      hence y * H = z * H by Th1,A7,A18;
    end;
    assume (y * H) * (x * H) = (z * H) * (x * H);
    then (y * x) * H = (z * H) * (x * H) by A8;
    then (y * x) * H = (z * x) * H by A8;
    then consider w such that
    A20: w in Nucl Q & z * x = (y * x) * w by A7;
    A21: w in Nucl_l Q & w in Nucl_r Q by Th12,A20;
    set v = (x * w) / x;
    A22: v in Nucl Q by A1,Th63,A20;
    then A23: v in Nucl_m Q by Th12;
    z * x = y * (v * x) by A20,Def24,A21
    .= (y * v) * x by Def23,A23;
    hence y * H = z * H by Th2,A7,A22;
  end;
  hence thesis;
end;

registration
  let Q be AIM multLoop;
  cluster lp (Nucl Q) -> normal;
  coherence by Th64;
end;

registration
  let Q be multLoop;
  cluster lp (Cent Q) -> normal;
  coherence by Th54;
end;

::$N Main Theorem The AIM Conjecture follows
::from knowing every AIM loop satisfies
::aa1, aa2, aa3, Ka, aK1, aK2 and aK3.
::This theorem justifies using first-order theorem provers
::to try to prove the AIM Conjecture.
theorem
  (for Q being multLoop st Q is satisfying_TT satisfying_TL satisfying_TR
     satisfying_LR satisfying_LL satisfying_RR
  holds Q is satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3)
  implies
  for Q being AIM multLoop holds
  Q _/_ (lp (Nucl Q)) is commutative multGroup
  &
  Q _/_ (lp (Cent Q)) is multGroup
proof
  assume A1: for Q being multLoop st Q is
     satisfying_TT satisfying_TL satisfying_TR satisfying_LR
     satisfying_LL satisfying_RR
  holds Q is satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3;
  let Q be AIM multLoop;
  reconsider Q1 = Q as satisfying_aa1 satisfying_aa2 satisfying_aa3
    satisfying_Ka satisfying_aK1 satisfying_aK2 satisfying_aK3 multLoop by A1;
  set NN = lp (Nucl Q);
  set fN = QuotientHom(Q,NN);
  A2: for y being Element of Q _/_ NN holds ex x being Element of Q st
    fN.x = y
  proof
    let y be Element of Q _/_ NN;
    y in Cosets NN;
    then consider x being Element of Q such that
    A3: y = x * NN by Def41;
    take x;
    thus thesis by A3,Def48;
  end;
    Ker (QuotientHom(Q,NN)) = @ ([#] NN) by Th44;
    then Nucl Q1 c= Ker fN by Th24;
  hence Q _/_ NN is commutative multGroup by Th16,A2;
  set NC = lp (Cent Q);
  set fC = QuotientHom(Q,NC);
  A4: for y being Element of Q _/_ NC holds ex x being Element of Q st
    fC.x = y
  proof
    let y be Element of Q _/_ NC;
    y in Cosets NC;
    then consider x being Element of Q such that
    A5: y = x * NC by Def41;
    fC.x = y by A5,Def48;
    hence thesis;
  end;
  Ker (QuotientHom(Q,NC)) = @ ([#] NC) by Th44;
  then Cent Q1 c= Ker fC by Th25;
  hence Q _/_ NC is multGroup by Th17,A4;
end;